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# 5.7: Solve Equations with Decimals

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Skills to Develop

• Determine whether a decimal is a solution of an equation
• Solve equations with decimals
• Translate to an equation and solve

be prepared!

Before you get started, take this readiness quiz.

1. Evaluate x + $$\dfrac{2}{3}$$ when x = $$− \dfrac{1}{4}$$. If you missed this problem, review Example 4.9.15.
2. Evaluate 15 − y when y = −5. If you missed this problem, review Example 3.6.12.
3. Solve $$\dfrac{n}{−7}$$ = 42. If you missed this problem, review Example 4.12.5.

## Determine Whether a Decimal is a Solution of an Equation

Solving equations with decimals is important in our everyday lives because money is usually written with decimals. When applications involve money, such as shopping for yourself, making your family’s budget, or planning for the future of your business, you’ll be solving equations with decimals.

Now that we’ve worked with decimals, we are ready to find solutions to equations involving decimals. The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, a fraction, or a decimal. We’ll list these steps here again for easy reference.

HOW TO: DETERMINE WHETHER A NUMBER IS A SOLUTION TO AN EQUATION

• Step 1. Substitute the number for the variable in the equation.
• Step 2. Simplify the expressions on both sides of the equation.
• Step 3. Determine whether the resulting equation is true.
• If so, the number is a solution.
• If not, the number is not a solution.

Example $$\PageIndex{1}$$:

Determine whether each of the following is a solution of x − 0.7 = 1.5: (a) x = 1 (b) x = −0.8 (c) x = 2.2

Solution

(a) x - 0.7 = 1.5

 Substitute $$\textcolor{red}{1}$$ for x. $$\textcolor{red}{1} - 0.7 \stackrel{?}{=} 1.5$$ Subtract. $$0.3 \neq 1.5$$

Since x = 1 does not result in a true equation, 1 is not a solution to the equation.

(b) x - 0.7 = 1.5

 Substitute $$\textcolor{red}{-0.8}$$ for x. $$\textcolor{red}{-0.8} - 0.7 \stackrel{?}{=} 1.5$$ Subtract. $$-1.5 \neq 1.5$$

Since x = −0.8 does not result in a true equation, −0.8 is not a solution to the equation.

(c) x - 0.7 = 1.5

 Substitute $$\textcolor{red}{2.2}$$ for x. $$\textcolor{red}{2.2} - 0.7 \stackrel{?}{=} 1.5$$ Subtract. $$1.5 = 1.5 \; \checkmark$$

Since x = 2.2 results in a true equation, 2.2 is a solution to the equation.

Exercise $$\PageIndex{1}$$:

Determine whether each value is a solution of the given equation. x − 0.6 = 1.3: (a) x = 0.7 (b) x = 1.9 (c) x = −0.7

no

yes

no

Exercise $$\PageIndex{2}$$:

Determine whether each value is a solution of the given equation. y − 0.4 = 1.7: (a) y = 2.1 (b) y = 1.3 (c) −1.3

yes

no

no

## Solve Equations with Decimals

In previous chapters, we solved equations using the Properties of Equality. We will use these same properties to solve equations with decimals.

Properties of Equality

 Subtraction Property of Equality For any numbers a, b, and c, If a = b, then a − c = b − c. Addition Property of Equality For any numbers a, b, and c, If a = b, then a + c = b + c. The Division Property of Equality For any numbers a, b, and c, and c ≠ 0 If a = b, then $$\dfrac{a}{c} = \dfrac{b}{c}$$ The Multiplication Property of Equality For any numbers a, b, and c, If a = b, then ac = bc

When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.

Example $$\PageIndex{2}$$:

Solve: y + 2.3 = −4.7.

Solution

We will use the Subtraction Property of Equality to isolate the variable.

 Subtract $$\textcolor{red}{2.3}$$ from each side, to undo the addition. $$y + 2.3 \textcolor{red}{-2.3} = -4.7 \textcolor{red}{-2.3}$$ Simplify. $$y = -7$$

Check:

 Substitute y = $$\textcolor{red}{-7}$$. $$\textcolor{red}{-7} + 2.3 \stackrel{?}{=} -4.7$$ Simplify. $$-4.7 = -4.7 \; \checkmark$$

Since y = −7 makes y + 2.3 = −4.7 a true statement, we know we have found a solution to this equation.

Exercise $$\PageIndex{3}$$:

Solve: y + 2.7 = −5.3.

$$y=-8$$

Exercise $$\PageIndex{4}$$:

Solve: y + 3.6 = −4.8.

$$y=-8.4$$

Example $$\PageIndex{3}$$:

Solve: a − 4.75 = −1.39.

Solution

We will use the Addition Property of Equality.

 Add 4.75 to each side, to undo the subtraction. $$a - 4.75 \textcolor{red}{+4.75} = -1.39 \textcolor{red}{+4.75}$$ Simplify. $$a = 3.36$$

Check:

 Substitute a = $$\textcolor{red}{3.36}$$. $$\textcolor{red}{3.36} - 4.75 \stackrel{?}{=} -1.39$$ $$-1.39 = -1.39 \; \checkmark$$

Since the result is a true statement, a = 3.36 is a solution to the equation.

Exercise $$\PageIndex{5}$$:

Solve: a − 3.93 = −2.86.

$$a=1.07$$

Exercise $$\PageIndex{6}$$:

Solve: n − 3.47 = −2.64.

$$n=0.83$$

Example $$\PageIndex{4}$$:

Solve: −4.8 = 0.8n.

Solution

We will use the Division Property of Equality. Use the Properties of Equality to find a value for n.

 We must divide both sides by 0.8 to isolate n. $$\dfrac{-4.8}{\textcolor{red}{0.8}} = \dfrac{0.8n}{\textcolor{red}{08}}$$ Simplify. $$-6 = n$$

Check:

 Substitute n = $$\textcolor{red}{-6}$$. $$-4.8 \stackrel{?}{=} 0.8 (\textcolor{red}{-6})$$ $$-4.8 = -4.8 \; \checkmark$$

Since n = −6 makes −4.8 = 0.8n a true statement, we know we have a solution.

Exercise $$\PageIndex{7}$$:

Solve: −8.4 = 0.7b.

$$b=-12$$

Exercise $$\PageIndex{8}$$:

Solve: −5.6 = 0.7c.

$$c=-8$$

Example $$\PageIndex{5}$$:

Solve: $$\dfrac{p}{−1.8}$$ = −6.5.

Solution

We will use the Multiplication Property of Equality.

 Here, p is divided by −1.8. We must multiply by −1.8 to isolate p. $$\textcolor{red}{-1.8} \left(\dfrac{p}{-1.8}\right) = \textcolor{red}{-1.8} (-6.5)$$ Multiply. $$p = 11.7$$

Check:

 Substitute p = $$\textcolor{red}{11.7}$$. $$\dfrac{\textcolor{red}{11.7}}{-1.8} \stackrel{?}{=} -6.5$$ $$-6.5 = -6.5 \; \checkmark$$

A solution to $$\dfrac{p}{−1.8}$$ = −6.5 is p = 11.7.

Exercise $$\PageIndex{9}$$:

Solve: $$\dfrac{c}{−2.6}$$ = −4.5.

$$c=-11.7$$

Exercise $$\PageIndex{10}$$:

Solve: $$\dfrac{b}{−1.2}$$ = −5.4.

$$b=6.48$$

## Translate to an Equation and Solve

Now that we have solved equations with decimals, we are ready to translate word sentences to equations and solve. Remember to look for words and phrases that indicate the operations to use.

Example $$\PageIndex{6}$$:

Translate and solve: The difference of n and 4.3 is 2.1.

Solution

 Translate. Add 4.3 to both sides of the equation. $$n - 4.3 \textcolor{red}{+4.3} = 2.1 \textcolor{red}{+4.3}$$ Simplify. $$n = 6.4$$

Check:

 Let n = 6.4. Is the difference of 6.4 and 4.3 equal to 2.1? Translate. $$6.4 - 4.3 \stackrel{?}{=} 2.1$$ Simplify. $$2.1 = 2.1 \; \checkmark$$

Exercise $$\PageIndex{11}$$:

Translate and solve: The difference of y and 4.9 is 2.8.

$$y-4.9=2.8$$; $$y=7.7$$

Exercise $$\PageIndex{12}$$:

Translate and solve: The difference of z and 5.7 is 3.4.

$$z-5.7=3.4$$; $$z=9.1$$

Example $$\PageIndex{7}$$:

Translate and solve: The product of −3.1 and x is 5.27.

Solution

 Translate. Divide both sides by −3.1. $$\dfrac{-3.1x}{\textcolor{red}{-3.1}} = \dfrac{5.27}{\textcolor{red}{-3.1}}$$ Simplify. $$x = -1.7$$

Check:

 Let x = −1.7. Is the product of −3.1 and −1.7 equal to 5.27? Translate. $$-3.1 (-1.7) \stackrel{?}{=} 5.27$$ Simplify. $$5.27 = 5.27 \; \checkmark$$

Exercise $$\PageIndex{13}$$:

Translate and solve: The product of −4.3 and x is 12.04.

$$-4.3x=12.04$$; $$x=-2.8$$

Exercise $$\PageIndex{14}$$:

Translate and solve: The product of −3.1 and m is 26.66.

$$-3.1m=26.66$$; $$m=-8.6$$

Example $$\PageIndex{8}$$:

Translate and solve: The quotient of p and −2.4 is 6.5.

Solution

 Translate. Multiply both sides by −2.4. $$\textcolor{red}{-2.4} \left(\dfrac{p}{-2.4}\right) = \textcolor{red}{-2.4} (6.5)$$ Simplify. $$p = -15.6$$

Check:

 Let p = −15.6. Is the quotient of −15.6 and −2.4 equal to 6.5? Translate. $$\dfrac{\textcolor{red}{-15.6}}{-2.4} \stackrel{?}{=} 6.5$$ Simplify. $$6.5 = 6.5 \; \checkmark$$

Exercise $$\PageIndex{15}$$:

Translate and solve: The quotient of q and −3.4 is 4.5.

$$\frac{1}{-3.4}=4.5$$; $$q=-15.3$$

Exercise $$\PageIndex{16}$$:

Translate and solve: The quotient of r and −2.6 is 2.5.

$$\frac{r}{-2.6}=2.5$$; $$r=-6.5$$

Example $$\PageIndex{9}$$:

Translate and solve: The sum of n and 2.9 is 1.7.

Solution

 Translate. Subtract 2.9 from each side. $$n + 2.9 \textcolor{red}{-2.9} = 1.7 \textcolor{red}{-2.9}$$ Simplify. $$n = -1.2$$

Check:

 Let n = −1.2. Is the sum −1.2 and 2.9 equal to 1.7? Translate. $$-1.2 + 2.9 \stackrel{?}{=} 1.7$$ Simplify. $$1.7 = 1.7 \; \checkmark$$

Exercise $$\PageIndex{17}$$:

Translate and solve: The sum of j and 3.8 is 2.6.

$$j+3.8=2.6$$; $$j=-1.2$$

Exercise $$\PageIndex{18}$$:

Translate and solve: The sum of k and 4.7 is 0.3.

$$k+4.7=0.3$$; $$k=-4.4$$

## Practice Makes Perfect

### Determine Whether a Decimal is a Solution of an Equation

In the following exercises, determine whether each number is a solution of the given equation.

1. x − 0.8 = 2.3
1. x = 2
2. x = −1.5
3. x = 3.1
2. y + 0.6 = −3.4
1. y = −4
2. y = −2.8
3. y = 2.6
3. $$\dfrac{h}{1.5}$$ = −4.3
1. h = 6.45
2. h = −6.45
3. h = −2.1
4. 0.75k = −3.6
1. k = −0.48
2. k = −4.8
3. k = −2.7

### Solve Equations with Decimals

In the following exercises, solve the equation.

1. y + 2.9 = 5.7
2. m + 4.6 = 6.5
3. f + 3.45 = 2.6
4. h + 4.37 = 3.5
5. a + 6.2 = −1.7
6. b + 5.8 = −2.3
7. c + 1.15 = −3.5
8. d + 2.35 = −4.8
9. n − 2.6 = 1.8
10. p − 3.6 = 1.7
11. x − 0.4 = −3.9
12. y − 0.6 = −4.5
13. j − 1.82 = −6.5
14. k − 3.19 = −4.6
15. m − 0.25 = −1.67
16. q − 0.47 = −1.53
17. 0.5x = 3.5
18. 0.4p = 9.2
19. −1.7c = 8.5
20. −2.9x = 5.8
21. −1.4p = −4.2
22. −2.8m = −8.4
23. −120 = 1.5q
24. −75 = 1.5y
25. 0.24x = 4.8
26. 0.18n = 5.4
27. −3.4z = −9.18
28. −2.7u = −9.72
29. $$\dfrac{a}{0.4}$$ = −20
30. $$\dfrac{b}{0.3}$$ = −9
31. $$\dfrac{x}{0.7}$$ = −0.4
32. $$\dfrac{y}{0.8}$$ = −0.7
33. $$\dfrac{p}{−5}$$ = −1.65
34. $$\dfrac{q}{−4}$$ = −5.92
35. $$\dfrac{r}{−1.2}$$ = −6
36. $$\dfrac{s}{−1.5}$$ = −3

### Mixed Practice

In the following exercises, solve the equation. Then check your solution.

1. x − 5 = −11
2. $$− \dfrac{2}{5}$$ = x + (\dfrac{3}{4}\)
3. p + 8 = −2
4. p + $$\dfrac{2}{3} = \dfrac{1}{12}$$
5. −4.2m = −33.6
6. q + 9.5 = −14
7. q + $$\dfrac{5}{6} = \dfrac{1}{12}$$
8. $$\dfrac{8.6}{15}$$ = − d
9. $$\dfrac{7}{8} m = \dfrac{1}{10}$$
10. $$\dfrac{j}{−6.2}$$ = −3
11. $$− \dfrac{2}{3}$$ = y + $$\dfrac{3}{8}$$
12. s − 1.75 = −3.2
13. $$\dfrac{11}{20}$$ = − f
14. −3.6b = 2.52
15. −4.2a = 3.36
16. −9.1n = −63.7
17. r − 1.25 = −2.7
18. $$\dfrac{1}{4} n = \dfrac{7}{10}$$
19. $$\dfrac{h}{−3}$$ = −8
20. y − 7.82 = −16

### Translate to an Equation and Solve

In the following exercises, translate and solve.

1. The difference of n and 1.9 is 3.4.
2. The difference n and 1.5 is 0.8.
3. The product of −6.2 and x is −4.96.
4. The product of −4.6 and x is −3.22.
5. The quotient of y and −1.7 is −5.
6. The quotient of z and −3.6 is 3.
7. The sum of n and −7.3 is 2.4.
8. The sum of n and −5.1 is 3.8.

## Everyday Math

1. Shawn bought a pair of shoes on sale for $78. Solve the equation 0.75p = 78 to find the original price of the shoes, p. 2. Mary bought a new refrigerator. The total price including sales tax was$1,350. Find the retail price, r, of the refrigerator before tax by solving the equation 1.08r = 1,350.

## Writing Exercises

1. Think about solving the equation 1.2y = 60, but do not actually solve it. Do you think the solution should be greater than 60 or less than 60? Explain your reasoning. Then solve the equation to see if your thinking was correct.
2. Think about solving the equation 0.8x = 200, but do not actually solve it. Do you think the solution should be greater than 200 or less than 200? Explain your reasoning. Then solve the equation to see if your thinking was correct.

## Self Check

(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

(b) On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?