11.3 Solving equations of the form x+a=b and x-a=b

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to solve equations of the form $x+a=b$ and $x-a=b$ . By the end of the module students should understand the meaning and function of an equation, understand what is meant by the solution to an equation and be able to solve equations of the form $x+a=b$ and $x-a=b$ .

Section overview

• Equations
• Solutions and Equivalent Equations
• Solving Equations

Equation

An equation is a statement that two algebraic expressions are equal.

The following are examples of equations:

$\underset{\text{expression}}{\underset{\text{This}}{\underbrace{x+6}}}\underset{=}{\underset{}{\underset{}{=}}}\underset{\text{expression}}{\underset{\text{This}}{\underbrace{10}}}$ $\underset{\text{expression}}{\underset{\text{This}}{\underbrace{x-4}}}\underset{=}{\underset{}{\underset{}{=}}}\underset{\text{expression}}{\underset{\text{This}}{\underbrace{-11}}}$ $\underset{\text{expression}}{\underset{\text{This}}{\underbrace{3y-5}}}\underset{=}{\underset{}{\underset{}{=}}}\underset{\text{expression}}{\underset{\text{This}}{\underbrace{-2+2y}}}$

Notice that $x+6$ , $x-4$ , and $3y-5$ are not equations. They are expressions. They are not equations because there is no statement that each of these expressions is equal to another expression.

Conditional equations

The truth of some equations is conditional upon the value chosen for the variable. Such equations are called conditional equations . There are two additional types of equations. They are examined in courses in algebra, so we will not consider them now.

Solutions and solving an equation

The set of values that, when substituted for the variables, make the equation true, are called the solutions of the equation.
An equation has been solved when all its solutions have been found.

Sample set a

Verify that 3 is a solution to $x+7=\text{10}$ .

When $x=3$ ,

Verify that $-6$ is a solution to $5y+8=-\text{22}$

When $y=-6$ ,

Verify that 5 is not a solution to $a-1=2a+3$ .

When $a=5$ ,

Verify that -2 is a solution to $3m-2=-4m-\text{16}$ .

When $m=-2$ ,

Practice set a

Verify that 5 is a solution to $m+6=\text{11}$ .

Substitute 5 into $m+6=\text{11}$ . Thus, 5 is a solution.

Verify that $-5$ is a solution to $2m-4=-\text{14}$ .

Substitute -5 into $2m-4=-\text{14}$ . Thus, -5 is a solution.

Verify that 0 is a solution to $5x+1=1$ .

Substitute 0 into $5x+1=1$ . Thus, 0 is a solution.

Verify that 3 is not a solution to $-3y+1=4y+5$ .

Substitute 3 into $-3y+1=4y+5$ . Thus, 3 is not a solution.

Verify that -1 is a solution to $6m-5+2m=7m-6$ .

Substitute -1 into $6m-5+2m=7m-6$ . Thus, -1 is a solution.

Equivalent equations

Some equations have precisely the same collection of solutions. Such equations are called equivalent equations. For example, $x-5=-1$ , $x+7=11$ , and $x=4$ are all equivalent equations since the only solution to each is $x=4$ . (Can you verify this?)

Solving equations

We know that the equal sign of an equation indicates that the number represented by the expression on the left side is the same as the number represented by the expression on the right side.

 This number is the same as this number ↓ ↓ ↓ $x$ = 4 $x+7$ = 11 $x-5$ = -1

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