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Solve: $\mathrm{-4.8}=0.8n.$
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 . | ||
Simplify. | ||
Check: | ||
Since $n=\mathrm{-6}$ makes $\mathrm{-4.8}=0.8n$ a true statement, we know we have a solution.
Solve: $\frac{p}{-1.8}=\mathrm{-6.5}.$
We will use the
Multiplication Property of Equality .
Here, p is divided by −1.8. We must multiply by −1.8 to isolate p | ||
Multiply. | ||
Check: | ||
A solution to $\frac{p}{\mathrm{-1.8}}=\mathrm{-6.5}$ is $p=11.7.$
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.
Translate and solve: The difference of $n$ and $4.3$ is $2.1.$
Translate. | ||
Add $4.3$ to both sides of the equation. | ||
Simplify. | ||
Check: | Is the difference of $n$ and 4.3 equal to 2.1? | |
Let $n=4.3$ : | Is the difference of 6.4 and 4.3 equal to 2.1? | |
Translate. | ||
Simplify. |
Translate and solve: The difference of $y$ and $4.9$ is $2.8.$
y − 4.9 = 2.8; y = 7.7
Translate and solve: The difference of $z$ and $5.7$ is $3.4.$
z − 5.7 = 3.4; z = 9.1
Translate and solve: The product of $\mathrm{-3.1}$ and $x$ is $5.27.$
Translate. | ||
Divide both sides by $\mathrm{-3.1}$ . | ||
Simplify. | ||
Check: | Is the product of −3.1 and $x$ equal to $5.27$ ? | |
Let $x=\mathrm{-1.7}$ : | Is the product of $\mathrm{-3.1}$ and $\mathrm{-1.7}$ equal to $5.27$ ? | |
Translate. | ||
Simplify. |
Translate and solve: The product of $\mathrm{-4.3}$ and $x$ is $12.04.$
−4.3 x = 12.04; x = −2.8
Translate and solve: The product of $\mathrm{-3.1}$ and $m$ is $26.66.$
−3.1 m = 26.66; m = −8.6
Translate and solve: The quotient of $p$ and $\mathrm{-2.4}$ is $6.5.$
Translate. | ||
Multiply both sides by $\mathrm{-2.4}$ . | ||
Simplify. | ||
Check: | Is the quotient of $p$ and $\mathrm{-2.4}$ equal to $6.5$ ? | |
Let $p=\mathrm{-15.6}:$ | Is the quotient of $\mathrm{-15.6}$ and $\mathrm{-2.4}$ equal to $6.5$ ? | |
Translate. | ||
Simplify. |
Translate and solve: The quotient of $q$ and $\mathrm{-3.4}$ is $4.5.$
$\frac{q}{\mathrm{-3.4}}=4.5;\phantom{\rule{0.2em}{0ex}}q=\mathrm{-15.3}$
Translate and solve: The quotient of $r$ and $\mathrm{-2.6}$ is $2.5.$
$\frac{r}{\mathrm{-2.6}}=2.5;\phantom{\rule{0.2em}{0ex}}r=\mathrm{-6.5}$
Translate and solve: The sum of $n$ and $2.9$ is $1.7.$
Translate. | ||
Subtract $2.9$ from each side. | ||
Simplify. | ||
Check: | Is the sum $n$ and $2.9$ equal to $1.7$ ? | |
Let $n=\mathrm{-1.2}:$ | Is the sum $\mathrm{-1.2}$ and $2.9$ equal to $1.7$ ? | |
Translate. | ||
Simplify. |
Translate and solve: The sum of $j$ and $3.8$ is $2.6.$
j + 3.8 = 2.6; j = −1.2
Translate and solve: The sum of $k$ and $4.7$ is $0.3.$
k + 4.7 = 0.3; k = −4.4
Subtraction Property of Equality | Addition Property of Equality |
For any numbers
$a$ ,
$b$ , and
$c$ ,
$\begin{array}{cccc}\text{If}& \hfill a& =& b\hfill \\ \text{then}& \hfill a-c& =& b-c\hfill \end{array}$ |
For any numbers
$a$ ,
$b$ , and
$c$ ,
$\begin{array}{cccc}\text{If}& \hfill a& =& b\hfill \\ \text{then}& \hfill a+c& =& b+c\hfill \end{array}$ |
Division of Property of Equality | Multiplication Property of Equality |
For any numbers
$a$ ,
$b$ , and
$c\ne 0$ ,
$\begin{array}{cccc}\text{If}& \hfill a& =& b\hfill \\ \text{then}& \hfill \frac{a}{c}& =& \frac{b}{c}\hfill \end{array}$ |
For any numbers
$a$ ,
$b$ , and
$c$ ,
$\begin{array}{cccc}\text{If}& \hfill a& =& b\hfill \\ \text{then}& \hfill a\cdot c& =& b\cdot c\hfill \end{array}$ |
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.
$x-0.8=2.3$
ⓐ
$\phantom{\rule{0.2em}{0ex}}x=2$
ⓑ
$\phantom{\rule{0.2em}{0ex}}x=\mathrm{-1.5}$
ⓒ
$\phantom{\rule{0.2em}{0ex}}x=3.1$
$y+0.6=\mathrm{-3.4}$
ⓐ
$\phantom{\rule{0.2em}{0ex}}y=\mathrm{-4}$
ⓑ
$\phantom{\rule{0.2em}{0ex}}y=\mathrm{-2.8}$
ⓒ
$\phantom{\rule{0.2em}{0ex}}y=2.6$
$\frac{h}{1.5}=\mathrm{-4.3}$
ⓐ
$\phantom{\rule{0.2em}{0ex}}h=6.45$
ⓑ
$\phantom{\rule{0.2em}{0ex}}h=\mathrm{-6.45}$
ⓒ
$\phantom{\rule{0.2em}{0ex}}h=\mathrm{-2.1}$
$0.75k=\mathrm{-3.6}$
ⓐ
$\phantom{\rule{0.2em}{0ex}}k=\mathrm{-0.48}$
ⓑ
$\phantom{\rule{0.2em}{0ex}}k=\mathrm{-4.8}$
ⓒ
$\phantom{\rule{0.2em}{0ex}}k=\mathrm{-2.7}$
Solve Equations with Decimals
In the following exercises, solve the equation.
$m+4.6=6.5$
$h+4.37=3.5$
$b+5.8=\mathrm{-2.3}$
$d+2.35=\mathrm{-4.8}$
$p-3.6=1.7$
$y-0.6=\mathrm{-4.5}$
$k-3.19=\mathrm{-4.6}$
$q-0.47=\mathrm{-1.53}$
$0.4p=9.2$
$\mathrm{-2.9}x=5.8$
$\mathrm{-2.8}m=\mathrm{-8.4}$
$\mathrm{-75}=1.5y$
$0.18n=5.4$
$\mathrm{-2.7}u=\mathrm{-9.72}$
$\frac{b}{0.3}=\mathrm{-9}$
$\frac{y}{0.8}=\mathrm{-0.7}$
$\frac{q}{-4}=\mathrm{-5.92}$
$\frac{s}{-1.5}=\mathrm{-3}$
Mixed Practice
In the following exercises, solve the equation. Then check your solution.
$-\frac{2}{5}=x+\frac{3}{4}$
$p+\frac{2}{3}=\frac{1}{12}$
$q+9.5=\mathrm{-14}$
$\frac{8.6}{15}=-d$
$\frac{j}{-6.2}=\mathrm{-3}$
$s-1.75=\mathrm{-3.2}$
$\mathrm{-3.6}b=2.52$
$\mathrm{-9.1}n=\mathrm{-63.7}$
$\frac{1}{4}n=\frac{7}{10}$
$y-7.82=\mathrm{-16}$
Translate to an Equation and Solve
In the following exercises, translate and solve.
The difference $n$ and $1.5$ is $0.8.$
The product of $\mathrm{-6.2}$ and $x$ is $\mathrm{-4.96}.$
−6.2 x = −4.96; 0.8
The product of $\mathrm{-4.6}$ and $x$ is $\mathrm{-3.22}.$
The quotient of $y$ and $\mathrm{-1.7}$ is $\mathrm{-5}.$
$\frac{y}{-1.7}=\mathrm{-5};\phantom{\rule{0.2em}{0ex}}8.5$
The quotient of $z$ and $\mathrm{-3.6}$ is $3.$
The sum of $n$ and $\mathrm{-7.3}$ is $\mathrm{2.4.}$
n + (−7.3) = 2.4; 9.7
The sum of $n$ and $\mathrm{-5.1}$ is $3.8.$
Shawn bought a pair of shoes on sale for $\mathrm{\$78}$ . Solve the equation $0.75p=78$ to find the original price of the shoes, $p.$
$104
Mary bought a new refrigerator. The total price including sales tax was $\text{\$1,350}.$ Find the retail price, $r,$ of the refrigerator before tax by solving the equation $1.08r=\mathrm{1,350}.$
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.
Answers will vary.
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.
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ 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?
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