CHAPTER 2.1

ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION

PROPERTIES OF EQUALITY

Determine whether the given number is a solution to the equation.

)4 71

2( 5

Substitute -½ for x.

Simplify

47 5

2

2 7 5

5 5

Right-hand side equals the left=hand side. Thus, -½ is a solution to the equation 4x + 7 = 5

4x + 7 = 5x = -½

Determine whether the given number is a solution to the equation.

46(3) 1 4 Substitute 3 for x.

Simplify

18 14 4

4 4 Right-hand side does not equal the left=hand side. Thus, 3 is not a solution to the equation -6x + 14 = 4

-6x + 14 = 4x = 3

If we have a linear equation we can “manipulate” it to get it in this form. We just need to make sure that whatever we do preserves the equality (keeps both sides =)

413 x We can add or subtract the same thing from both sides of the equation.- 4 - 4

- 3 0Notice this is the equation above where a = 3 and b = -3.While this is in the general form for a linear

equation, we often want to find all values of x so that the equation is true. You could probably do this one in your head and see that when x = 1 we’d have a true statement 0 = 0

0 baxA linear equation in one variable is equivalent to an equation of the form:

033 x

ADDITION AND SUBTRACTION PROPERTIES OF EQUALITY

Let a, b, and c represent algebraic expressionsAddition property of equality: If

a = b,

then a + c = b + c

Subtraction property of equality: If a = b,

then a – c = b -c

APPLYING THE ADDITION AND SUBTRACTION PROPERTIES OF EQUALITY

In each equation, the goal is to isolate the variable on one side of the equation. To accomplish this, we use the fact that the sum of a number and its opposite is zero and the difference of a number and itself is zero.

p – 4 +4 = 4 +4

To isolate p, add 4 to both sides (-4 +4 = 0).

p – 4 = 11

Simplifyp- + 0 = 15p = 15

CHECK

MULTIPLICATION AND DIVISION PROPERTIES OF EQUALITY

Multiplication and Division Properties of EqualityLet a, b, and c represent algebraic expressions

1. Multiplication property of equality: If a = b, then ac = bc

2. Division property of equality: If a = b

then

provided c ≠ 0

a

c c

b

Applying the Multiplication and Division Properties of Equality

12x = 60Tip: Recall that the product of a number and its reciprocal is 1. For example:

(112

12) 1

12x = 60 12 12

To obtain a coefficient of 1 for the x-term, divide both sides by 12

Simplify

x = 5

Check!

Example:

2 1

9 3q

Tip: When applying the multiplication or division properties of equality to obtain a coefficient of 1 for the variable term, we will generally use the following convention:1. If the coefficient of the variable term is

expressed as a fraction, we usually multiply both sides by its reciprocal.

2. If the coefficient of the variable term is an integer or decimal, we divide both sides by the coefficient itself.

29 9( ) ( )

2( )92

1

3q To obtain a coefficient of 1 for

the q-term, multiply by the reciprocal ofwhich is

2

9

9

2

Simplify. The product of a number and its reciprocal is 1.

3

2q

CHECK!

Translating to a Linear Equation

The quotient of a number and 4 is 6The product of a number and 4 is 6

Negative twelve is equal to the sum of -5 and a number

The value 1.4 subtracted from a number is 5.7