11.3: Solving Equations of the Form x + a = b and x - a = b

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Learning Objectives

understand the meaning and function of an equation
understand what is meant by the solution to an equation
be able to solve equations of the form $x + a = b$ and $x - a = b$

Equations

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

The following are examples of equations:

$\begin{array} {c} {\underbrace{x + 6}_{\text{This}}} & = & {\underbrace{10}_{\text{This}}} \\ {^\text{expression}} & ^= & {^\text{expression}} \end{array}$ $\begin{array} {c} {\underbrace{x - 4}_{\text{This}}} & = & {\underbrace{-11}_{\text{This}}} \\ {^\text{expression}} & ^= & {^\text{expression}} \end{array}$ $\begin{array} {c} {\underbrace{3y - 5}_{\text{This}}} & = & {\underbrace{-2 + 2y}_{\text{This}}} \\ {^\text{expression}} & ^= & {^\text{expression}} \end{array}$

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.

Solutions and Equivalent Equations

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 $x + 7 = 10$.

Solution

When $x = 3$,

becomes $\begin{array} {rcll} {x + 7} & = & {10} & {} \\ {3 + 7} & = & {10} & {} \\ {10} & = & {10} & {\text{which is a } true \text{ statement, verifying that}} \\ {} & & {} & {\text{3 is a solution to } x + 7 = 10} \end{array}$

Sample Set A

Verify that -6 is a solution to $5y + 8 = -22$.

Solution

When $y = -6$,

becomes $\begin{array} {rcll} {5y + 8} & = & {-22} & {} \\ {5(-6) + 8} & = & {-22} & {} \\ {-30 + 8} & = & {-22} & {} \\ {-22} & = & {-22} & {\text{which is a } true \text{ statement, verifying that}} \\ {} & & {} & {\text{-6 is a solution to } 5y + 8 = -22} \end{array}$

Sample Set A

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

Solution

When $a = 5$,

becomes $\begin{array} {rcll} {a - 1} & = & {2a + 3} & {} \\ {5 - 1} & = & {2 \cdot 5 + 3} & {} \\ {5 - 1} & = & {10 + 3} & {} \\ {4} & = & {13} & {\text{a } false \text{ statement, verifying that 5}} \\ {} & & {} & {\text{is not a solution to } a - 1 = 2a + 3} \end{array}$

Sample Set A

Verify that -2 is a solution to $3m - 2 = -4m - 16$.

Solution

When $m = -2$,

becomes $\begin{array} {rcll} {3m - 2} & = & {-4m - 16} & {} \\ {3(-2) - 2} & = & {-4(-2) - 16} & {} \\ {-6 - 2} & = & {8 - 16} & {} \\ {-8} & = & {-8} & {\text{which is a } true \text{ statement, verifying that}} \\ {} & & {} & {\text{-2 is a solution to } 3m - 2 = -4m - 16} \end{array}$

Practice Set A

Verify that 5 is a solution to $m + 6 = 11$.

Answer

Substitute 5 into $m + 6 = 11$.

$Does 5 plus 6 equal 11? Yes.$

Thus, 5 is a solution.

Practice Set A

Verify that −5 is a solution to $2m - 4 = -14$.

Answer

Substitute -5 into $2m - 4 = -14$.

$does 2 time negative 5 minus 4 equal negative 14? Yes.$

Thus, -5 is a solution.

Practice Set A

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

Answer

Substitute 0 into $5x + 1 = 1$.

$Does 5 times zero plus one equal 1? Yes.$

Thus, 0 is a solution.

Practice Set A

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

Answer

Substitute 3 into $-3y + 1 = 4y + 5$.

$Does negative 3 times 3 plus 1 equal 4 times 3 plus 5? No.$

Thus, 3 is not a solution.

Practice Set A

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

Answer

Substitute -1 into $6m - 5 + 2m = 7m - 6$.

$Does 6 times negative 1 minus 5 plus 2 times negative 1 equal 7 times negative 1 minus 6? Yes.$

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
$\downarrow$	$\downarrow$	$\downarrow$
$x$	=	4
$x + 7$	=	11
$x - 5$	=	-1

Addition/Subtraction Property of Equality
From this, we can suggest the addition/subtraction property of equality.
Given any equation,

We can obtain an equivalent equation by adding the same number to both sides of the equation.
We can obtain an equivalent equation by subtracting the same number from both sides of the equation.

The Idea Behind Equation Solving
The idea behind equation solving is to isolate the variable on one side of the equation. Signs of operation (+, -, $\cdot$, $\div$) are used to associate two numbers. For example, in the expression $5 + 3$, the numbers 5 and 3 are associated by addition. An association can be undone by performing the opposite operation. The addition/subtraction property of equality can be used to undo an association that is made by addition or subtraction.

Subtraction is used to undo an addition.

Addition is used to undo a subtraction.

The procedure is illustrated in the problems of [link].

Sample Set B

Use the addition/subtraction property of equality to solve each equation.

$x + 4 = 6$.

Solution

4 is associated with $x$ by addition. Undo the association by subtracting 4 from both sides.

$x + 4 - 4 = 6 - 4$

$x + 0 =2$

$x = 2$

Check: When $x = 2$, $x + 4$ becomes

$Does 2 + 4 = 6? Yes.$

The solution to $x + 4 = 6$ is $x = 2$.

Sample Set B

$m - 8 = 5$.

Solution

8 is associated with $m$ by subtraction. Undo the association by adding 8 to both sides.

$m - 8 + 8 = 5 + 8$

$m + 0 = 13$

$m = 13$

Check: When $m = 13$,

becomes

$m - 8 = 5. After substituting, does 13 - 8 = 5? Yes.$
a true statement.

The solution to $m - 8 = 5$ is $m = 13$.

Sample Set B

$-3 - 5 = y - 2 + 8$.

Solution

Before we use the addition/subtraction property, we should simplify as much as possible.

$-3 - 5 = y - 2 + 8$.

$-8 = y + 6$

6 is associated with $y$ by addition. Undo the association by subtracting 6 from both sides.

$-8 - 6 = y + 6 - 6$

$-14 = y + 0$

$-14 = y$

This is equivalent to $y = -14$.

Check: When $y = -14$,

$-3 - 5 = y - 2 + 8$

becomes

$Does negative 3 minus 5 equal negative 14 minus 2 plus 8? Yes.$

a true statement.

The solution to $-3 - 5 = y - 2 + 8$ is $y = -14$.

Sample Set B

$-5a + 1 + 6a = -2$.

Solution

Begin by simplifying the left side of the equation.

$\underbrace{-5a + 1 + 6a}_{-5 + 6 = 1} = -2$

$a + 1 = -2$ 1 is associated with aa by addition. Undo the association by subtracting 1 from both sides.

$a + 1 - 1 = -2 - 1$

$a + 0 = -3$

$a = -3$

Check: When $a = -3$,

$-5a + 1 + 6a = -2$

becomes

$Does negative 5 times negative 3 plus 1 plus 6 times negative 3 equal negative 2? Yes.$
a true statement.

The solution to $-5a + 1 + 6a = -2$ is $a = -3$.

Sample Set B

$7k - 4 = 6k + 1$.

Solution

In this equation, the variable appears on both sides. We need to isolate it on one side. Although we can choose either side, it will be more convenient to choose the side with the larger coefficient. Since 8 is greater than 6, we’ll isolate $k$ on the left side.

$7 k - 4 = 6k + 1$ Since $6k$ represents $+6k$, subtract $6k$ from each side.

$\underbrace{7 k - 4 - 6k}_{7 - 6 = 1} = \underbrace{6k + 1 - 6k}_{6 - 6 = 0}$

$k - 4 = 1$ 4 is associated with $k$ by subtraction. Undo the association by adding 4 to both sides.

\(k - 4 + 4 = 1 + 4\0

$k = 5$

Check: When $k = 5$.

$7k - 4 = 6k + 1$

becomes

$Does 7 times 5 minus 4 equal 6 times 5 plus 1? Yes.$

a true statement.

The solution to $7k - 4 = 6k + 1$ is $k = 5$

Sample Set B

$-8 + x = 5$.

Solution

-8 is associated with $x$ by addition. Undo the by subtracting -8 from both sides. Subtracting -8 we get $-(-8) = +8$. We actually add 8 to both sides.

$-8 + x + 8 = 5 + 8$

$x = 13$

Check: When $x = 13$

$-8 + x = 5$

becomes

$Does negative 8 plus 13 equal 5? Yes.$

a true statement.

The solution to $-8 + x = 5$ is $x = 13$.

Practice Set B

$y + 9 = 4$

Answer: $y = -5$

Practice Set B

$a - 4 = 11$

Answer: $a = 15$

Practice Set B

$-1 + 7 = x + 3$

Answer: $x = 3$

Practice Set B

$8m + 4 - 7m = (-2) (-3)$

Answer: $m = 2$

Practice Set B

$12k - 4 = 9k - 6 + 2k$

Answer: $k = -2$

Practice Set B

$-3 + a = -4$

Answer: $a = -1$

Exercises

For the following 10 problems, verify that each given value is a solution to the given equation.

Exercise $\PageIndex{1}$

$x - 11 = 5$, $x = 16$

Answer

Substitue $x = 4$ into the equation $4x - 11 = 5$.

$16 - 11 = 5$

$5 = 5$

$x = 4$ is a solution

Exercise $\PageIndex{2}$

$y - 4 = -6$, $y = -2$

Exercise $\PageIndex{3}$

$2m - 1 = 1$, $m = 1$

Answer

Substitue $m = 1$ into the equation $2m - 1 = 1$.

$Does 2 minus 1 equal 1? Yes.$

$m = 1$ is a solution.

Exercise $\PageIndex{4}$

$5y + 6 = -14$, $y = -4$

Exercise $\PageIndex{5}$

$3x + 2 - 7x = -5x - 6$, $x = -8$

Answer

Substitue $x = -8$ into the equation $3x + 2 - 7 = -5x - 6$.

$Does negative 24 plus 2 minus 7 equal 40 minus 6? Yes.$

$x = -8$ is a solution.

Exercise $\PageIndex{6}$

$-6a + 3 + 3a = 4a + 7 - 3a$, $a = -1$

Exercise $\PageIndex{7}$

$-8 + x = -8$, $x = 0$

Answer

Substitue $x = 0$ into the equation $-8 + x = -8$.

$Does negative 8 plus 0 equal negative 8? Yes.$

$x = 0$ is a solution.

Exercise $\PageIndex{8}$

$8b + 6 = 6 - 5b$, $b = 0$

Exercise $\PageIndex{9}$

$4x - 5 = 6x - 20$, $x = \dfrac{15}{2}$

Answer

Substitue $x = \dfrac{15}{2}$ into the equation $4x - 5 = 6x - 20$.

$Does 30 minus 5 equal 45 minus 20? Yes.$

$x = \dfrac{15}{2}$ is a solution

Exercise $\PageIndex{10}$

$-3y + 7 = 2y - 15$, $y = \dfrac{22}{5}$

Solve each equation. Be sure to check each result.

Exercise $\PageIndex{11}$

$y - 6 = 5$

Answer: $y = 11$

Exercise $\PageIndex{12}$

$m + 8 = 4$

Exercise $\PageIndex{13}$

$k - 1 = 4$

Answer: $k = 5$

Exercise $\PageIndex{14}$

$h - 9 = 1$

Exercise $\PageIndex{15}$

$a + 5 = -4$

Answer: $a = -9$

Exercise $\PageIndex{16}$

$b - 7 = -1$

Exercise $\PageIndex{17}$

$x + 4 - 9 = 6$

Answer: $x = 11$

Exercise $\PageIndex{18}$

$y - 8 + 10 = 2$

Exercise $\PageIndex{19}$

$z + 6 = 6$

Answer: $z = 0$

Exercise $\PageIndex{20}$

$w - 4 = -4$

Exercise $\PageIndex{21}$

$x + 7 - 9 = 6$

Answer: $x = 8$

Exercise $\PageIndex{22}$

$y - 2 + 5 = 4$

Exercise $\PageIndex{23}$

$m + 3 - 8 = -6 + 2$

Answer: $m = 1$

Exercise $\PageIndex{24}$

$z + 10 - 8 = -8 + 10$

Exercise $\PageIndex{25}$

$2 + 9 = k - 8$

Answer: $k = 19$

Exercise $\PageIndex{26}$

$-5 + 3 = h - 4$

Exercise $\PageIndex{27}$

$3m - 4 = 2m + 6$

Answer: $m = 10$

Exercise $\PageIndex{28}$

$5a + 6 = 4a - 8$

Exercise $\PageIndex{29}$

$8b + 6 + 2b = 3b - 7 + 6b - 8$

Answer: $b = -21$

Exercise $\PageIndex{30}$

$12h - 1 - 3 - 5h = 2h + 5h + 3(-4)$

Exercise $\PageIndex{31}$

$-4a + 5 - 2a = -3a - 11 - 2a$

Answer: $a = 16$

Exercise $\PageIndex{32}$

$-9n - 2 - 6 + 5n = 3n - (2) (-5) - 6n$

Calculator Exercises

Exercise $\PageIndex{33}$

$y - 2.161 = 6.063$

Answer: $y = 7.224$

Exercise $\PageIndex{34}$

$a - 44.0014 = -21.1625$

Exercise $\PageIndex{35}$

$-0.362 - 0.416 = 5.63m - 4.63m$

Answer: $m = -0.778$

Exercise $\PageIndex{36}$

$8.078 - 9.112 = 2.106y - 1.106y$

Exercise $\PageIndex{37}$

$4.23k + 3.18 = 3.23k - 5.83$

Answer: $k = -9.01$

Exercise $\PageIndex{38}$

$6.1185x - 4.0031 = 5.1185x - 0.0058$

Exercise $\PageIndex{39}$

$21.63y + 12.40 - 5.09y = 6.11y - 15.66 + 9.43y$

Answer: $y = -28.06$

Exercise $\PageIndex{40}$

$0.029a - 0.013 - 0.034 -0.057 = -0.038 + 0.56 + 1.01a$

Exercises for Review

Exercise $\PageIndex{41}$

Is $\dfrac{\text{7 calculators}}{\text{12 students}}$ an example of a ratio or a rate?

Answer: rate

Exercise $\PageIndex{42}$

Convert $\dfrac{3}{8}$% to a decimal

Exercise $\PageIndex{43}$

0.4% of what number is 0.014?

Answer: 3.5

Exercise $\PageIndex{44}$

Use the clustering method to estimate the sum: $89 + 93 + 206 + 198 + 91$

Exercise $\PageIndex{45}$

Combine like terms: $4x + 8y + 12y + 9x - 2y$.

Answer: $13x + 18y$

This number	is the same as	this number
\(\downarrow\)	\(\downarrow\)	\(\downarrow\)
\(x\)	=	4
\(x + 7\)	=	11
\(x - 5\)	=	-1

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Solutions and Equivalent Equations

Solving Equations

Exercises for Review