1     Arithmetic Properties

An equation is like a balance scale because it shows that two quantities are equal. As with scales, the two sides of an equation remain balanced when the same amount is added to or taken away from each side.

To solve an equation containing a variable, you find the value or values of the variable that make the equation true. Such a value is a solution of the equation. To find a solution, you can use properties of equality to form equivalent equations, which are equations that have the same solution or solutions.

Of course, to use the properties of equality you must know what they are. Let us first look at the

4 + 2     6

1.0

1.0

Arithmetic Properties

Solving Equations

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Key Concepts

Solve x – 4 = – 9

Check your solution in the original equation by substituting – 5 for x

=

=

=

– 9

– 9 + 4

– 5

x – 4

  x – 4 + 4

              x

Add 4 to each side to get the variable

alone on one side of the equal sign.

Simplify.

x – 4

  – 5 – 4

              – 9

=

=

=

– 9

– 9

– 9

Using the Addition Property of Equality

EXAMPLE

This is where we start getting into the heart of Algebra I, solving equations to solve real-life problems. On the next few pages we will be looking specifically at linear equations and their solutions. We’ll start off slow and solve equations that use only one property to make sure you have the individual concepts down. Later we’ll pick up the pace and solve equations where you need to use several properties and steps to get the job done.

Remember that one way to solve an equation is to get the variable alone on one side of the equal sign, and that you can do this using inverse operations -- operations that undo one another -- such as addition and subtraction (or you might think in terms of a positive undoing a negative and vice versa).

When you solve an equation involving subtraction (a negative sign), add the same number to each side of the equation.

Using the Subtraction Property of Equality

EXAMPLE

Again, one way to solve an equation is to get the variable alone on one side of the equal sign by using inverse operations such as addition and subtraction (or  positive and negative).

So, when you solve an equation involving addition, subtract the same number from each side of the equation.

The Addition and Subtraction Properties of Equality

If a = b, then a + c = b + c

If a = b, then a - c = b - c

Solve x – 4 = – 9

Check your solution in the original equation by substituting – 5 for x

=

=

=

– 9

– 9 + 4

– 5

x – 4

  x – 4 + 4

              x

Add 4 to each side to get the variable

alone on one side of the equal sign.

Simplify.

x – 4

  – 5 – 4

              – 9

=

=

=

– 9

– 9

– 9

REVIEW

EXAMPLE

Solve –      = – 9

Check your solution in the original equation by substituting 36 for x.

=

=

=

– 9

– 4(– 9 )

36

–   

  – 4( –      )

              x

Multiply each side by – 4 to get the variable alone on one side of the equal sign.

Simplify.

–   

  –    

              – 9

=

=

=

– 9

– 9

– 9

x

4

x

4

x

4

x

4

36

4

Using the Multiplication Property of Equality

Multiplication and division are inverse operations. When you solve an equation involving division, multiply each side of the equation by the same number.

– 48

– 48

–12

Solve     4x = – 48

Check your solution in the original equation by substituting –12 for x

=

=

=

4x

4x

              x

Divide each side by 4.

Simplify.

4x

  4(–12)

              – 48

=

=

=

– 48

– 48

– 48

___        ___

  4           4

Again, multiplication and division are inverse operations. So, when you solve an equation involving multiplication, divide each side of the equation by the same number.

EXAMPLE

Using the Division Property of Equality

When the coefficient of the variable is a fraction, you can use reciprocals to solve the equation.

Solve     x = 9

Check your solution in the original equation by substituting 12 for x

4   EXAMPLE      Using Reciprocals to Solve Equations

=

=

=

9

  (9)

12

x

(    x)

              x

Multiply each side by 4/3, the reciprocal of 3/4.

Simplify.

x

  (12)

              9

=

=

=

9

9

9

3

4

3

4

3

4

4

3

4

3

3

4

3

4

2

Property        Division Property of Equality

For every real number a, b, and c, with c ≠ 0, if a = b, then a/c = b/c

Property        Multiplication Property of Equality

For every real number a, b, and c, if a = b, then a · c = b · c

Example ¼ = 0.25, so ¼ · 5 = 0.25 · 5

Property        Subtraction Property of Equality

For every real number a, b, and c, if a = b, then a – c = b – c

Example 10 = 6 + 4, so 10 – 5 = (6 + 4) – 5

Property        Addition Property of Equality

For every real number a, b, and c, if a = b, then a + c = b + c

Example 10 = 6 + 4, so 10 + 5 = (6 + 4) + 5

Example 4 + 2 = 6, so

=

2

2