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### CHAPTER 2.1

ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION PROPERTIES OF EQUALITY

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

4x + 7 = 5

x = -½

Substitute -½ for x.

Simplify

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

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

-6x + 14 = 4

x = 3

Substitute 3 for x.

Simplify

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

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

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 =)

We can add or subtract the same thing from both sides of the equation.

- 4

- 4

- 3

0

Notice 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

ADDITION AND SUBTRACTION PROPERTIES OF EQUALITY equation of the form:

- Let a, b, and c represent algebraic expressions
- Addition 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 equation of the form:

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 = 11

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

p – 4 +4 = 4 +4

p- + 0 = 15

p = 15

Simplify

CHECK

MULTIPLICATION AND DIVISION PROPERTIES OF EQUALITY equation of the form:

- Multiplication and Division Properties of Equality
- Let a, b, and c represent algebraic expressions
- 1. Multiplication property of equality: If a = b, then ac = bc
- Division property of equality: If a = b
- then
- provided c ≠ 0

Applying the Multiplication and Division Properties of Equality

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

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

12x = 60

12x = 60

12 12

Simplify

x = 5

Check!

- 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:
- If the coefficient of the variable term is expressed as a fraction, we usually multiply both sides by its reciprocal.
- If the coefficient of the variable term is an integer or decimal, we divide both sides by the coefficient itself.

Example:

To obtain a coefficient of 1 for the q-term, multiply by the reciprocal of

which is

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

CHECK!

Translating to a Linear Equation properties of equality to obtain a coefficient of 1 for the variable term, we will generally use the following convention:

The quotient of a number and 4 is 6

The 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

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