Algebra Worksheets

Welcome to the Algebra worksheets page at Math-Drills.com, where unknowns are common and variables are the norm. On this page, you will find Algebra worksheets for middle school students on topics such as algebraic expressions, equations and graphing functions.

This page starts off with some missing numbers worksheets for younger students. We then get right into algebra by helping students recognize and understand the basic language related to algebra. The rest of the page covers some of the main topics you'll encounter in algebra units. Remember that by teaching students algebra, you are helping to create the future financial whizzes, engineers, and scientists that will solve all of our world's problems.

Algebra is much more interesting when things are more real. Solving linear equations is much more fun with a two pan balance, some mystery bags and a bunch of jelly beans. Algebra tiles are used by many teachers to help students understand a variety of algebra topics. And there is nothing like a set of co-ordinate axes to solve systems of linear equations.

Most Popular Algebra Worksheets this Week

Solving Quadratic Equations with Positive or Negative 'a' Coefficients up to 4

Algebraic Properties, Rules and Laws Worksheets

algebra properties worksheet answer key

The commutative law or commutative property states that you can change the order of the numbers in an arithmetic problem and still get the same results. In the context of arithmetic, it only works with addition or multiplication operations , but not mixed addition and multiplication. For example, 3 + 5 = 5 + 3 and 9 × 5 = 5 × 9. A fun activity that you can use in the classroom is to brainstorm non-numerical things from everyday life that are commutative and non-commutative. Putting on socks, for example, is commutative because you can put on the right sock then the left sock or you can put on the left sock then the right sock and you will end up with the same result. Putting on underwear and pants, however, is non-commutative.

  • The Commutative Law Worksheets The Commutative Law of Addition (Numbers Only) The Commutative Law of Addition (Some Variables) The Commutative Law of Multiplication (Numbers Only) The Commutative Law of Multiplication (Some Variables)

The associative law or associative property allows you to change the grouping of the operations in an arithmetic problem with two or more steps without changing the result. The order of the numbers stays the same in the associative law. As with the commutative law, it applies to addition-only or multiplication-only problems. It is best thought of in the context of order of operations as it requires that parentheses must be dealt with first. An example of the associative law is: (9 + 5) + 6 = 9 + (5 + 6). In this case, it doesn't matter if you add 9 + 5 first or 5 + 6 first, you will end up with the same result. Students might think of some examples from their experience such as putting items on a tray at lunch. They could put the milk and vegetables on their tray first then the sandwich or they could start with the vegetables and sandwich then put on the milk. If their tray looks the same both times, they will have modeled the associative law. Reading a book could be argued as either associative or nonassociative as one could potentially read the final chapters first and still understand the book as well as someone who read the book the normal way.

  • The Associative Law Worksheets The Associative Law of Addition (Whole Numbers Only) The Associative Law of Multiplication (Whole Numbers Only)

Inverse relationships worksheets cover a pre-algebra skill meant to help students understand the relationship between multiplication and division and the relationship between addition and subtraction.

  • Inverse Mathematical Relationships with One Blank Addition and Subtraction Easy Addition and Subtraction Harder All Multiplication and Division Facts 1 to 18 in color (no blanks) Multiplication and Division Range 1 to 9 Multiplication and Division Range 5 to 12 Multiplication and Division All Inverse Relationships Range 2 to 9 Multiplication and Division All Inverse Relationships Range 5 to 12 Multiplication and Division All Inverse Relationships Range 10 to 25
  • Inverse Mathematical Relationships with Two Blanks Addition and Subtraction (Sums 1-18) Addition and Subtraction Inverse Relationships with 1 Addition and Subtraction Inverse Relationships with 2 Addition and Subtraction Inverse Relationships with 3 Addition and Subtraction Inverse Relationships with 4 Addition and Subtraction Inverse Relationships with 5 Addition and Subtraction Inverse Relationships with 6 Addition and Subtraction Inverse Relationships with 7 Addition and Subtraction Inverse Relationships with 8 Addition and Subtraction Inverse Relationships with 9 Addition and Subtraction Inverse Relationships with 10 Addition and Subtraction Inverse Relationships with 11 Addition and Subtraction Inverse Relationships with 12 Addition and Subtraction Inverse Relationships with 13 Addition and Subtraction Inverse Relationships with 14 Addition and Subtraction Inverse Relationships with 15 Addition and Subtraction Inverse Relationships with 16 Addition and Subtraction Inverse Relationships with 17 Addition and Subtraction Inverse Relationships with 18

The distributive property is an important skill to have in algebra. In simple terms, it means that you can split one of the factors in multiplication into addends, multiply each addend separately, add the results, and you will end up with the same answer. It is also useful in mental math, an example of which should help illustrate the definition. Consider the question, 35 × 12. Splitting the 12 into 10 + 2 gives us an opportunity to complete the question mentally using the distributive property. First multiply 35 × 10 to get 350. Second, multiply 35 × 2 to get 70. Lastly, add 350 + 70 to get 420. In algebra, the distributive property becomes useful in cases where one cannot easily add the other factor before multiplying. For example, in the expression, 3(x + 5), x + 5 cannot be added without knowing the value of x. Instead, the distributive property can be used to multiply 3 × x and 3 × 5 to get 3x + 15.

  • Distributive Property Worksheets Distributive Property (Answers do not include exponents) Distributive Property (Some answers include exponents) Distributive Property (All answers include exponents)

Students should be able to substitute known values in for an unknown(s) in an expression and evaluate the expression's value.

  • Evaluating Expressions with Known Values Evaluating Expressions with One Variable, One Step and No Exponents Evaluating Expressions with One Variable and One Step Evaluating Expressions with One Variable and Two Steps Evaluating Expressions with Up to Two Variables and Two Steps Evaluating Expressions with Up to Two Variables and Three Steps Evaluating Expressions with Up to Three Variables and Four Steps Evaluating Expressions with Up to Three Variables and Five Steps

As the title says, these worksheets include only basic exponent rules questions. Each question only has two exponents to deal with; complicated mixed up terms and things that a more advanced student might work out are left alone. For example, 4 2 is (2 2 ) 2 = 2 4 , but these worksheets just leave it as 4 2 , so students can focus on learning how to multiply and divide exponents more or less in isolation.

  • Exponent Rules for Multiplying, Dividing and Powers Mixed Exponent Rules (All Positive) Mixed Exponent Rules (With Negatives) Multiplying Exponents (All Positive) Multiplying Exponents (With Negatives) Multiplying the Same Exponent with Different Bases (All Positive) Multiplying the Same Exponent with Different Bases (With Negatives) Dividing Exponents with a Greater Exponent in Dividend (All Positive) Dividing Exponents with a Greater Exponent in Dividend (With Negatives) Dividing Exponents with a Greater Exponent in Divisor (All Positive) Dividing Exponents with a Greater Exponent in Divisor (With Negatives) Powers of Exponents (All Positive) Powers of Exponents (With Negatives)

Knowing the language of algebra can help to extract meaning from word problems and to situations outside of school. In these worksheets, students are challenged to convert phrases into algebraic expressions.

  • Translating Algebraic Phrases into Expressions Translating Algebraic Phrases into Expressions (Simple Version) Translating Algebraic Phrases into Expressions (Complex Version)

Combining like terms is something that happens a lot in algebra. Students can be introduced to the topic and practice a bit with these worksheets. The bar is raised with the adding and subtracting versions that introduce parentheses into the expressions. For students who have a good grasp of fractions, simplifying simple algebraic fractions worksheets present a bit of a challenge over the other worksheets in this section.

  • Simplifying Expressions by Combining Like Terms Simplifying Linear Expressions with 3 terms Simplifying Linear Expressions with 4 terms Simplifying Linear Expressions with 5 terms Simplifying Linear Expressions with 6 to 10 terms
  • Simplifying Expressions by Combining Like Terms with Some Arithmetic Adding and simplifying linear expressions Adding and simplifying linear expressions with multipliers Adding and simplifying linear expressions with some multipliers . Subtracting and simplifying linear expressions Subtracting and simplifying linear expressions with multipliers Subtracting and simplifying linear expressions with some multipliers Mixed adding and subtracting and simplifying linear expressions Mixed adding and subtracting and simplifying linear expressions with multipliers Mixed adding and subtracting and simplifying linear expressions with some multipliers Simplify simple algebraic fractions (easier) Simplify simple algebraic fractions (harder)
  • Rewriting Linear Equations Rewrite Linear Equations in Standard Form Convert Linear Equations from Standard to Slope-Intercept Form Convert Linear Equations from Slope-Intercept to Standard Form Convert Linear Equations Between Standard and Slope-Intercept Form
  • Rewriting Formulas Rewriting Formulas (addition and subtraction; about one step) Rewriting Formulas (addition and subtraction; about two steps) Rewriting Formulas ( multiplication and division ; about one step)

Linear Expressions and Equations

algebra properties worksheet answer key

In these worksheets, the unknown is limited to the question side of the equation which could be on the left or the right of equal sign.

  • Missing Numbers in Equations with Blanks as Unknowns Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Blanks in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 20 ; Blanks in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 20 ; Blanks in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 20 ; Blanks in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 20 ; Blanks in Any Position )
  • Missing Numbers in Equations with Symbols as Unknowns Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Symbols in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 20 ; Symbols in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 20 ; Symbols in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 20 ; Symbols in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 20 ; Symbols in Any Position )
  • Solving Equations with Addition and Symbols as Unknowns Equalities with Addition (0 to 9) Symbol Unknowns Equalities with Addition (1 to 12) Symbol Unknowns Equalities with Addition (1 to 15) Symbol Unknowns Equalities with Addition (1 to 25) Symbol Unknowns Equalities with Addition (1 to 99) Symbol Unknowns
  • Missing Numbers in Equations with Variables as Unknowns Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Variables Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Variables in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 20 ; Variables in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 20 ; Variables in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 20 ; Variables in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 20 ; Variables in Any Position )
  • Solving Simple Linear Equations Solving Simple Linear Equations with Values from -9 to 9 (Unknown on Left Side) Solving Simple Linear Equations with Values from -99 to 99 (Unknown on Left Side) Solving Simple Linear Equations with Values from -9 to 9 (Unknown on Right or Left Side) Solving Simple Linear Equations with Values from -99 to 99 (Unknown on Right or Left Side)
  • Determining Linear Equations from Slopes, y-intercepts and Points Determine a Linear Equation from the Slope and y-intercept Determine a Linear Equation from the Slope and a Point Determine a Linear Equation from Two Points Determine a Linear Equation from Two Points by Graphing

Graphing linear equations and reading existing graphs give students a visual representation that is very useful in understanding the concepts of slope and y-intercept.

  • Graphing Linear Equations Graph Slope-Intercept Equations
  • Determinging Linear Equations from Graphs Determine the Equation from a Graph Determine the Slope from a Graph Determine the y-intercept from a Graph Determine the x-intercept from a Graph Determine the slope and y-intercept from a Graph Determine the slope and intercepts from a Graph Determine the slope, intercepts and equation from a Graph

Solving linear equations with jelly beans is a fun activity to try with students first learning algebraic concepts. Ideally, you will want some opaque bags with no mass, but since that isn't quite possible (the no mass part), there is a bit of a condition here that will actually help students understand equations better. Any bags that you use have to be balanced on the other side of the equation with empty ones.

Probably the best way to illustrate this is through an example. Let's use 3 x + 2 = 14. You may recognize the x as the unknown which is actually the number of jelly beans we put in each opaque bag. The 3 in the 3 x means that we need three bags. It's best to fill the bags with the required number of jelly beans out of view of the students, so they actually have to solve the equation.

On one side of the two-pan balance, place the three bags with x jelly beans in each one and two loose jelly beans to represent the + 2 part of the equation. On the other side of the balance, place 14 jelly beans and three empty bags which you will note are required to "balance" the equation properly. Now comes the fun part... if students remove the two loose jelly beans from one side of the equation, things become unbalanced, so they need to remove two jelly beans from the other side of the balance to keep things even. Eating the jelly beans is optional. The goal is to isolate the bags on one side of the balance without any loose jelly beans while still balancing the equation.

The last step is to divide the loose jelly beans on one side of the equation into the same number of groups as there are bags. This will probably give you a good indication of how many jelly beans there are in each bag. If not, eat some and try again. Now, we realize this won't work for every linear equation as it is hard to have negative jelly beans, but it is another teaching strategy that you can use for algebra.

Despite all appearances, equations of the type a/ x are not linear. Instead, they belong to a different kind of equations. They are good for combining them with linear equations, since they introduce the concept of valid and invalid answers for an equation (what will be later called the domain of a function). In this case, the invalid answers for equations in the form a/ x , are those that make the denominator become 0.

  • Solving Linear Equations Combining Like Terms and Solving Simple Linear Equations Solving a x = c Linear Equations Solving a x = c Linear Equations including negatives Solving x /a = c Linear Equations Solving x /a = c Linear Equations including negatives Solving a/ x = c Linear Equations Solving a/ x = c Linear Equations including negatives Solving a x + b = c Linear Equations Solving a x + b = c Linear Equations including negatives Solving a x - b = c Linear Equations Solving a x - b = c Linear Equations including negatives Solving a x ± b = c Linear Equations Solving a x ± b = c Linear Equations including negatives Solving x /a ± b = c Linear Equations Solving x /a ± b = c Linear Equations including negatives Solving a/ x ± b = c Linear Equations Solving a/ x ± b = c Linear Equations including negatives Solving various a/ x ± b = c and x /a ± b = c Linear Equations Solving various a/ x ± b = c and x /a ± b = c Linear Equations including negatives Solving linear equations of all types Solving linear equations of all types including negatives

Linear Systems

algebra properties worksheet answer key

  • Solving Systems of Linear Equations Easy Linear Systems with Two Variables Easy Linear Systems with Two Variables including negative values Linear Systems with Two Variables Linear Systems with Two Variables including negative values Easy Linear Systems with Three Variables; Easy Easy Linear Systems with Three Variables including negative values Linear Systems with Three Variables Linear Systems with Three Variables including negative values
  • Solving Systems of Linear Equations by Graphing Solve Linear Systems by Graphing (Solutions in first quadrant only) Solve Standard Linear Systems by Graphing Solve Slope-Intercept Linear Systems by Graphing Solve Various Linear Systems by Graphing Identify the Dependent Linear System by Graphing Identify the Inconsistent Linear System by Graphing

Quadratic Expressions and Equations

algebra properties worksheet answer key

  • Simplifying (Combining Like Terms) Quadratic Expressions Simplifying quadratic expressions with 5 terms Simplifying quadratic expressions with 6 terms Simplifying quadratic expressions with 7 terms Simplifying quadratic expressions with 8 terms Simplifying quadratic expressions with 9 terms Simplifying quadratic expressions with 10 terms Simplifying quadratic expressions with 5 to 10 terms
  • Adding/Subtracting and Simplifying Quadratic Expressions Adding and simplifying quadratic expressions. Adding and simplifying quadratic expressions with multipliers. Adding and simplifying quadratic expressions with some multipliers. Subtracting and simplifying quadratic expressions. Subtracting and simplifying quadratic expressions with multipliers. Subtracting and simplifying quadratic expressions with some multipliers. Mixed adding and subtracting and simplifying quadratic expressions. Mixed adding and subtracting and simplifying quadratic expressions with multipliers. Mixed adding and subtracting and simplifying quadratic expressions with some multipliers.
  • Multiplying Factors to Get Quadratic Expressions Multiplying Factors of Quadratics with Coefficients of 1 Multiplying Factors of Quadratics with Coefficients of 1 or -1 Multiplying Factors of Quadratics with Coefficients of 1, or 2 Multiplying Factors of Quadratics with Coefficients of 1, -1, 2 or -2 Multiplying Factors of Quadratics with Coefficients up to 9 Multiplying Factors of Quadratics with Coefficients between -9 and 9

The factoring quadratic expressions worksheets in this section provide many practice questions for students to hone their factoring strategies. If you would rather worksheets with quadratic equations, please see the next section. These worksheets come in a variety of levels with the easier ones are at the beginning. The 'a' coefficients referred to below are the coefficients of the x 2 term as in the general quadratic expression: ax 2 + bx + c. There are also worksheets in this section for calculating sum and product and for determining the operands for sum and product pairs.

  • Factoring Quadratic Expressions Factoring Quadratic Expressions with Positive 'a' coefficients of 1 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients of 1 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients of 1 with a Common Factor Step Factoring Quadratic Expressions with Positive 'a' coefficients up to 4 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 4 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 4 with a Common Factor Step Factoring Quadratic Expressions with Positive 'a' coefficients up to 5 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 5 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 5 with a Common Factor Step Factoring Quadratic Expressions with Positive 'a' coefficients up to 9 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 9 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 9 with a Common Factor Step Factoring Quadratic Expressions with Positive 'a' coefficients up to 81 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 81 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 81 with a Common Factor Step Calculating Sum and Product (Operand Range 0 to 9 ) ✎ Calculating Sum and Product (Operand Range 1 to 9 ) ✎ Calculating Sum and Product (Operand Range 0 to 9 Including Negatives ) ✎ Calculating Sum and Product (Operand Range 1 to 9 Including Negatives ) ✎ Calculating Sum and Product (Operand Range -20 to 20 ) ✎ Calculating Sum and Product (Operand Range -99 to 99 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 0 to 9 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 1 to 9 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 0 to 12 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 1 to 12 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 0 to 9 Including Negatives ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 1 to 9 Including Negatives ) ✎ Determining Operands from Sum and Product Pairs (Operand Range -20 to 20 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range -99 to 99 ) ✎

Whether you use trial and error, completing the square or the general quadratic formula, these worksheets include a plethora of practice questions with answers. In the first section, the worksheets include questions where the quadratic expressions equal 0. This makes the process similar to factoring quadratic expressions, with the additional step of finding the values for x when the expression is equal to 0. In the second section, the expressions are generally equal to something other than x, so there is an additional step at the beginning to make the quadratic expression equal zero.

  • Solving Quadratic Equations that Equal Zero Solving Quadratic Equations with Positive 'a' coefficients of 1 Solving Quadratic Equations with Positive or Negative 'a' coefficients of 1 Solving Quadratic Equations with Positive or Negative 'a' coefficients of 1 with a Common Factor Step Solving Quadratic Equations with Positive 'a' coefficients up to 4 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 4 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 4 with a Common Factor Step Solving Quadratic Equations with Positive 'a' coefficients up to 5 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 5 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 5 with a Common Factor Step Solving Quadratic Equations with Positive 'a' coefficients up to 9 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 9 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 9 with a Common Factor Step Solving Quadratic Equations with Positive 'a' coefficients up to 81 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 81 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 81 with a Common Factor Step
  • Solving Quadratic Equations that Equal an Integer Solving Quadratic Equations for x ("a" coefficients of 1) Solving Quadratic Equations for x ("a" coefficients of 1 or -1) Solving Quadratic Equations for x ("a" coefficients up to 4) Solving Quadratic Equations for x ("a" coefficients between -4 and 4) Solving Quadratic Equations for x ("a" coefficients up to 81) Solving Quadratic Equations for x ("a" coefficients between -81 and 81)

Other Polynomial and Monomial Expressions & Equations

algebra properties worksheet answer key

  • Simplifying Polynomials That Involve Addition And Subtraction Addition and Subtraction; 1 variable; 3 terms Addition and Subtraction; 1 variable; 4 terms Addition and Subtraction; 2 variables; 4 terms Addition and Subtraction; 2 variables; 5 terms Addition and Subtraction; 2 variables; 6 terms
  • Simplifying Polynomials That Involve Multiplication And Division Multiplication and Division; 1 variable; 3 terms Multiplication and Division; 1 variable; 4 terms Multiplication and Division; 2 variables; 4 terms Multiplication and Division; 2 variables; 5 terms
  • Simplifying Polynomials That Involve Addition, Subtraction, Multiplication And Division All Operations; 1 variable; 3 terms All Operations; 1 variable; 4 terms All Operations; 2 variables; 4 terms All Operations; 2 variables; 5 terms All Operations (Challenge)
  • Factoring Expressions That Do Not Include A Squared Variable Factoring Non-Quadratic Expressions with No Squares, Simple Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with No Squares, Simple Coefficients, and Negative and Positive Multipliers Factoring Non-Quadratic Expressions with No Squares, Compound Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with No Squares, Compound Coefficients, and Negative and Positive Multipliers
  • Factoring Expressions That Always Include A Squared Variable Factoring Non-Quadratic Expressions with All Squares, Simple Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with All Squares, Simple Coefficients, and Negative and Positive Multipliers Factoring Non-Quadratic Expressions with All Squares, Compound Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with All Squares, Compound Coefficients, and Negative and Positive Multipliers
  • Factoring Expressions That Sometimes Include Squared Variables Factoring Non-Quadratic Expressions with Some Squares, Simple Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with Some Squares, Simple Coefficients, and Negative and Positive Multipliers Factoring Non-Quadratic Expressions with Some Squares, Compound Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with Some Squares, Compound Coefficients, and Negative and Positive Multipliers
  • Multiplying Polynomials With Two Factors Multiplying a monomial by a binomial Multiplying two binomials Multiplying a monomial by a trinomial Multiplying a binomial by a trinomial Multiplying two trinomials Multiplying two random mon/polynomials
  • Multiplying Polynomials With Three Factors Multiplying a monomial by two binomials Multiplying three binomials Multiplying two binomials by a trinomial Multiplying a binomial by two trinomials Multiplying three trinomials Multiplying three random mon/polynomials

Inequalities

algebra properties worksheet answer key

  • Writing The Inequality That Matches The Graph Writing Inequalities for Graphs
  • Graphing Inequalities On Number Lines Graphing Inequalities (Basic)
  • Solving Linear Inequalities Solving Inequalities Including a Third Term Solving Inequalities Including a Third Term and Multiplication Solving Inequalities Including a Third Term, Multiplication and Division

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Chapter 1: Algebra Review

1.4 Properties of Algebra (Review)

When doing algebra, it is common not to know the value of the variables. In this case, simplify where possible and leave any unknown variables in the final solution. One way to simplify expressions is to combine like terms.

Like terms are terms whose variables match exactly, exponents included. Examples of like terms would be [latex]3xy[/latex] and [latex]-7xy,[/latex] [latex]3a^2b[/latex] and [latex]8a^2b,[/latex] or −3 and 5. To combine like terms, add (or subtract) the numbers in front of the variables and keep the variables the same.

Example 1.4.1

Simplify [latex]5x - 2y - 8x + 7y.[/latex]

[latex]\begin{array}{rl} 5x - 8x \text{ and } -2y + 7y & \text{Combine like terms} \\ \\ -3x + 5y & \text{Solution} \end{array}[/latex]

Example 1.4.2

Simplify [latex]8x^2 - 3x + 7 - 2x^2 + 4x - 3.[/latex]

[latex]\begin{array}{rl} 8x^2 - 2x^2, -3x + 4x, \text{ and } 7 - 3 & \text{Combine like terms} \\ \\ 6x^2 + x + 4 & \text{Solution} \end{array}[/latex]

When combining like terms, subtraction signs must be interpreted as part of the terms they precede. This means that the term following a subtraction sign should be treated like a negative term. The sign always stays with the term.

Another method to simplify is known as distributing. Sometimes, when working with problems, there will be a set of parentheses that makes solving a problem difficult, if not impossible. To get rid of these unwanted parentheses, use the distributive property and multiply the number in front of the parentheses by each term inside.

[latex]\text{Distributive Property: } a(b + c) = ab + ac[/latex]

Several examples of using the distributive property are given below.

Example 1.4.3

Simplify [latex]4(2x-7).[/latex]

[latex]\begin{array}{rl} 4(2x-7)& \text{Multiply each term by } 4. \\ \\ 8x-28 & \text{Solution} \end{array}[/latex]

Example 1.4.4

Simplify [latex]-7(5x-6).[/latex]

[latex]\begin{array}{rl} -7(5x-6) & \text{Multiply each term by }-7. \\ \\ -35x+42 & \text{Solution} \end{array}[/latex]

In the previous example, it is necessary to again use the fact that the sign goes with the number. This means −6 is treated as a negative number, which gives (−7)(−6) = 42, a positive number. The most common error in distributing is a sign error. Be very careful with signs! It is possible to distribute just a negative throughout parentheses. If there is a negative sign in front of parentheses, think of it like a −1 in front and distribute it throughout.

Example 1.4.5

Simplify [latex]-(4x-5y+6).[/latex]

[latex]\begin{array}{rl} -(4x-5y+6) & \text{Negative can be thought of as }-1. \\ \\ -1(4x-5y+6) & \text{Multiply each term by }-1. \\ \\ -4x+5y-6 & \text{Solution} \end{array}[/latex]

Distributing throughout parentheses and combining like terms can be combined into one problem. Order of operations says to multiply (distribute) first, then add or subtract (combine like terms). Thus, do each problem in two steps: distribute, then combine.

Example 1.4.6

Simplify [latex]3x-2(4x-5).[/latex]

[latex]\begin{array}{rl} 3x-2(4x-5) & \text{Distribute }-2, \text{ multiplying each term.} \\ \\ 3x-8x+10 & \text{Combine like terms }3x-8x. \\ \\ -5x+10 & \text{Solution} \end{array}[/latex]

Example 1.4.7

Simplify [latex]5+3(2x-4).[/latex]

[latex]\begin{array}{rl} 5+3(2x-4) & \text{Distribute 3, multiplying each term.} \\ \\ 5+6x-12 & \text{Combine like terms }5-12. \\ \\ -7+6x & \text{Solution} \end{array}[/latex]

In Example 1.4.6, −2 is distributed, not just 2. This is because a number being subtracted must always be treated like it has a negative sign attached to it. This makes a big difference, for in that example, when the −5 inside the parentheses is multiplied by −2, the result is a positive number. More involved examples of distributing and combining like terms follow.

Example 1.4.8

Simplify [latex]2(5x-8)-6(4x+3).[/latex]

[latex]\begin{array}{rl} 2(5x-8)-6(4x+3) & \text{Distribute 2 into the first set of parentheses and }-6\text{ into the second.} \\ \\ 10x-16-24x-18 & \text{Combine like terms }10x-24x\text{ and }-16-18. \\ \\ -14x-34 & \text{Solution} \end{array}[/latex]

Example 1.4.9

Simplify [latex]4(3x-8)-(2x-7).[/latex]

[latex]\begin{array}{rl} 4(3x-8)-(2x-7) & \text{The negative sign in the middle can be thought of as }-1. \\ \\ 4(3x-8)-(2x-7) & \text{Distribute 4 into the first set of parentheses and }-1\text{ into the second.} \\ \\ 12x-32-2x+7 & \text{Combine like terms }12x-2x\text{ and }-32+7. \\ \\ 10x-25& \text{Solution} \end{array}[/latex]

For questions 1 to 28, reduce and combine like terms.

  • [latex]r - 9 + 10[/latex]
  • [latex]-4x + 2 - 4[/latex]
  • [latex]n + n[/latex]
  • [latex]4b + 6 + 1 + 7b[/latex]
  • [latex]8v + 7v[/latex]
  • [latex]-x + 8x[/latex]
  • [latex]-7x - 2x[/latex]
  • [latex]-7a - 6 + 5[/latex]
  • [latex]k - 2 + 7[/latex]
  • [latex]-8p + 5p[/latex]
  •  [latex]x - 10 - 6x + 1[/latex]
  • [latex]1 - 10n - 10[/latex]
  • [latex]m - 2m[/latex]
  • [latex]1 - r - 6[/latex]
  • [latex]-8(x - 4)[/latex]
  • [latex]3(8v + 9)[/latex]
  • [latex]8n(n + 9)[/latex]
  • [latex]-(-5 + 9a)[/latex]
  • [latex]7k(-k + 6)[/latex]
  • [latex]10x(1 + 2x)[/latex]
  • [latex]-6(1 + 6x)[/latex]
  • [latex]-2(n + 1)[/latex]
  • [latex]8m(5 - m)[/latex]
  • [latex]-2p(9p - 1)[/latex]
  • [latex]-9x(4 - x)[/latex]
  • [latex]4(8n - 2)[/latex]
  • [latex]-9b(b - 10)[/latex]
  • [latex]-4(1 + 7r)[/latex]

For questions 29 to 58, simplify each expression.

  • [latex]9(b + 10) + 5b[/latex]
  • [latex]4v - 7(1 - 8v)[/latex]
  • [latex]-3x(1 - 4x) - 4x^2[/latex]
  • [latex]-8x + 9(-9x + 9)[/latex]
  • [latex]-4{k}^{2} - 8k(8k + 1)[/latex]
  • [latex]-9 - 10(1 + 9a)[/latex]
  • [latex]1 - 7(5 + 7p)[/latex]
  • [latex]-10(x - 2) - 3[/latex]
  • [latex]-10 - 4(n - 5)[/latex]
  • [latex]-6(5 - m) + 3m[/latex]
  • [latex]4(x + 7) + 8(x + 4)[/latex]
  • [latex]-2r(1 + 4r) + 8r(-r + 4)[/latex]
  • [latex]-8(n + 6) - 8n(n + 8)[/latex]
  • [latex]9(6b + 5) - 4b(b + 3)[/latex]
  • [latex]7(7 + 3v) + 10(3 - 10v)[/latex]
  • [latex]-7(4x - 6) + 2(10x - 10)[/latex]
  • [latex]2n(- 10n + 5) - 7(6 - 10n)[/latex]
  • [latex]-3(4 + a) + 6a(9a + 10)[/latex]
  • [latex]5(1 - 6k) + 10(k - 8)[/latex]
  • [latex]-7(4x + 3) - 10(10x + 10)[/latex]
  • [latex](8n^2 - 3n) - (5 + 4n^2)[/latex]
  • [latex](7{x}^{2} - 3) - (5{x}^{2} + 6x)[/latex]
  • [latex](5p - 6) + (1 - p)[/latex]
  • [latex](3x^2 - x) - (7 - 8x)[/latex]
  • [latex](2 - 4v^2) + (3v^2 + 2v)[/latex]
  • [latex](2b - 8) + (b - 7b^2)[/latex]
  • [latex](4 - 2{k}^{2}) + (8 - 2{k}^{2})[/latex]
  • [latex](7{a}^{2} + 7a) - (6{a}^{2} + 4a)[/latex]
  • [latex]({x}^{2} - 8) + (2{x}^{2} - 7)[/latex]
  • [latex](3 - 7n^2) + (6n^2 + 3)[/latex]

Answer Key 1.4

Intermediate Algebra by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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1-4 Properties of Real Numbers

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  • Writing variable expressions
  • Order of operations
  • Evaluating expressions
  • Number sets
  • Adding rational numbers
  • Adding and subtracting rational numbers
  • Multiplying and dividing rational numbers
  • The distributive property
  • Combining like terms
  • Percent of change
  • One-step equations
  • Two-step equations
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  • Absolute value equations
  • Solving proportions
  • Percent problems
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  • Discrete relations
  • Continuous relations
  • Evaluating and graphing functions
  • Finding slope from a graph
  • Finding slope from two points
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  • Graphing lines using slope-intercept form
  • Graphing lines using standard form
  • Writing linear equations
  • Graphing linear inequalities
  • Graphing absolute value equations
  • Direct variation
  • Solving systems of equations by graphing
  • Solving systems of equations by elimination
  • Solving systems of equations by substitution
  • Systems of equations word problems
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  • Discrete exponential growth and decay word problems
  • Exponential functions and graphs
  • Writing numbers in scientific notation
  • Operations with scientific notation
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  • Naming polynomials
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  • Multiplying polynomials
  • Multiplying special case polynomials
  • Factoring special case polynomials
  • Factoring by grouping
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  • Graphing quadratic inequalities
  • Completing the square
  • By taking square roots
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  • Simplifying radicals
  • Adding and subtracting radical expressions
  • Multiplying radicals
  • Dividing radicals
  • Using the distance formula
  • Using the midpoint formula
  • Simplifying rational expressions
  • Finding excluded values / restricted values
  • Multiplying rational expressions
  • Dividing rational expressions
  • Adding and subtracting rational expressions
  • Finding trig. ratios
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  • Center and spread of data
  • Scatter plots
  • Using statistical models

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Number Properties Worksheets

What Are the Commutative, Associative, and Distributive Properties of Numbers? The commutative, associative and distributive properties are the underlying laws of algebra. They are first taught in early years in simpler and broader terms. Teaching these properties in early years gives a firm foundation to the children, which later on help them in understanding the advanced concepts. Here we have elaborated on the three properties of numbers. Commutative Property - Commutative is derived from the word commute, which means to move around. The commutative property means moving the numbers around within the number sentence. In algebraic terms, this commutative property implies being able to move the variables around. The equation of commutative property of addition defines: a + b = b + a, 3 + 4 = 4 + 3. The equation of the commutative property of multiplication defines: a x b = b x a, 3 x 4 = 4 x 3. Associative Property - The associative property is derived from the word associate or group. It means the grouping of variables and numbers. You can regroup or rearrange the variables in the number sentence but still, get the same answer. The equation of associative property of addition says: (a + b) + c = a + (b + c), (4 + 2) + 1 = 4 + (2 + 1). The equation of associative property of multiplication says: (a x b) x c = a x (b x c), (4 x 2) x 1 = 4 x (2 x 1). Distributive Property - Distributive property allows you to eliminate the brackets in the number expression. Multiply the values outside the parenthesis with each term present in the brackets. The equation of distributive property can be written as: a x (b + c) = a x b + a x c.

Basic Lesson

Demonstrates the commutative, associative, and distributive properties. Identify the property in the equation. Practice problems are provided.

Intermediate Lesson

Explains how to the property displayed in an equation. Practice problems are provided. Identify the property in the equation. z (d-c) = zd – zc.

Independent Practice 1

Write the name the number property used in each problem. The answers can be found below.

Independent Practice 2

Features another 20 Number Properties problems.

Homework Worksheet

12 Number Properties problems for students to work on at home. Example problems are provided and explained.

10 Number Properties problems. A math scoring matrix is included.

Homework and Quiz Answer Key

Answers for the homework and quiz.

Lesson and Practice Answer Key

Answers for both lessons and both practice sheets.

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