PreAlgebra

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Chapter 8 Expressions and Equations

8-7 Solving Equations

Definition

An equation is a mathematical statement of equality between two expressions.

Example: 2x + 3 = 11 is an equation

Note

An equal sign is used for indicating equality between two expressions.

Solving Equations of the Form a + b = c and a – b = c

Here is an example of an equation of form a – b = c

x – 2 = 8

The strategy for solving this equation is to do whatever is necessary to make variable x stand alone on the left side of this equal sign. This means getting rid of the – 2 term. And how do we make a – 2 go away? We add +2 to it: – 2 + 2 = 0. That's zero ... the – 2 is gone, and the x variable stands alone on the left side of the equation. But remember the cardinal rule of algebra: Whatever is done to one side of the equal sign must also be done to the other side. In this example, we added 2 to the left side of the equal sign, so we must follow this by adding 2 to the right side.

x – 2 + 2 = 8 + 2

Then we clean up the equation by combining like terms:

x – 2 + 2 = 8 + 2
x + 0 = 10

The result is x = 10

Example

Given: y – 5 = 12
Find: y

Problem

Solve y – 5 = 12

Add 5 to both sides of the equaton

y – 5 + 5 = 12 + 5

Combine like terms

y = 17

Solution

y = 17

Check

Does y – 5 = 12 when y = 17?

y – 5 = 12
17 – 5 = 12
12 = 12

Yes, it checks

Here is an example of an equation of form a + b = c

x + 2 = 8

The strategy for solving this equation is to do whatever is necessary to make variable x stand alone on the left side of this equal sign. This means getting rid of the + 2 term. And how do we make a + 2 go away? We subtract 2 from it: 2 – 2 = 0. That's zero ... the 2 is gone, and the x variable stands alone on the left side of the equation. But remember the cardinal rule of algebra: Whatever is done to one side of the equal sign must also be done to the other side. In this example, we subtracted 2 from the left side, so we must follow this by subtracting 2 from the right side.

x + 2 – 2 = 8 – 2

Then we clean up the equation by combining like terms:

x + 2 – 2 = 8 – 2
x + 0 = 6

The result is x = 6

Example

Given: y + 5 = 12
Find: y

Problem

Solve y + 5 = 12

Subtract 5 from both sides of the equaton

y + 5 – 5 = 12 – 5

Combine like terms

y = 7

Solution

y = 7

Check

Does y – 5 = 12 when y = 17?

y + 5 = 12
7+ 5 = 12
12 = 12

Yes, it checks

Exercises

Click the ? symbol to see the correct answer.

1. = ?	2. = ?	3. = ?	4. = ?	5. = ?
6. = ?	7. = ?	8. = ?	9. = ?	10. = ?

Solving Equations of Forms ax = b and x / a = b

Here is an example of an equation of form ax = c

2x = 8

Thinking Mathematically

Recall that any term divided by iteslf is equal to 1. Expressed as an equation, this is:

^a/_a = 1

Recall that any term multiplied by 1 is equal to that term, itself. Expressed as an equation:

a · 1 = a

The strategy for solving equations of the form ax = b uses both of these rules.

Here is an example of an equation of form x / a = b

^x/₂ = 8

Thinking Mathematically

Recall that any term multiplied by 1 is equal to that term, itself. Expressed as an equation:

a · 1 = a

Recall that any term divided by iteslf is equal to 1. Expressed as an equation, this is:

^a/_a = 1

The strategy for solving equations of the form x / a = b uses both of these rules.

Author: David L. Heiserman
Publisher: SweetHaven Publishing Services