In SectionÂ 3.1 we covered the steps to solve a linear equation and the differences among simplifying expressions, evaluating expressions and solving equations.

To solve this equation, we will simplify each side of the equation, manipulate it so that all variable terms are on one side and all constant terms are on the other, and then solve for \(a\text{:}\)

In SectionÂ 3.5 we covered the definitions of a ratio and a proportion and how to solve a proportion. We learned about cross multiplication, did problems about similar triangles, and used proportions to solve word problems.

Example3.7.5

Solve \(\frac{6-x}{5}=\frac{x}{4}\) for \(x\text{.}\)

So, the solution set is \(\left\{\frac{24}{9}\right\}\text{.}\)

Example3.7.6

Property taxes for a residential property are proportional to the assessed value of the property. Assume that a certain property in a given neighborhood is assessed at \(\$234{,}100\) and its annual property taxes are \(\$2{,}518.92\text{.}\) What are the annual property taxes for a house that is assessed at \(\$287{,}500\text{?}\)

The property taxes for a property assessed at \(\$287{,}500\) are \(\$3{,}093.50\text{.}\)

Example3.7.7

Since the two triangles are similar, we know that their side length should be proportional. To determine the unknown length, we can set up a proportion and solve for \(x\text{:}\)

\begin{align*}
\frac{\text{bigger triangle's left side length in cm}}{\text{bigger triangle's bottom side length in cm}}\amp=\frac{\text{smaller triangle's left side length in cm}}{\text{smaller triangle's bottom side length in cm}}\\
\frac{x\,\text{cm}}{6\,\text{cm}}\amp=\frac{3\,\text{cm}}{4\,\text{cm}}\\
\frac{x}{6}\amp=\frac{3}{4}\\
\multiplyleft{12}\frac{x}{6}\amp=\multiplyleft{12}\frac{3}{4}\qquad\text{(12 is the least common denominator)}\\
2x\amp=9\\
\divideunder{2x}{2}\amp=\divideunder{9}{2}\\
x\amp=\frac{9}{2}\ \text{or}\ 4.5
\end{align*}

The unknown side length is then 4.5âŻcm.

Subsection3.7.6Special Solution Sets

In SectionÂ 3.6 we covered linear equations that have no solutions and also linear equations that have infinitely many solutions. When solving linear inequalities, it's also possible that no solution exists or that all real numbers are solutions.

Example3.7.9

Solve for \(x\) in the equation \(3x=3x+4\text{.}\)

Solve for \(t\) in the inequality \(4t+5\gt 4t+2\text{.}\)

All values of the variable \(t\) make the inequality true. The solution set is all real numbers, which we can write as \(\{t\mid t\text{ is a real number}\}\) in set notation, or \((-\infty,\infty)\) in interval notation.

Carly has \({\$70}\) in her piggy bank. She plans to purchase some Pokemon cards, which costs \({\$1.15}\) each. She plans to save \({\$52.75}\) to purchase another toy. At most how many Pokemon cards can he purchase?

Write an equation to solve this problem.

Carly can purchase at most Pokemon cards.

26

Maygen has \({\$72}\) in her piggy bank. She plans to purchase some Pokemon cards, which costs \({\$2.55}\) each. She plans to save \({\$43.95}\) to purchase another toy. At most how many Pokemon cards can he purchase?

Write an equation to solve this problem.

Maygen can purchase at most Pokemon cards.

27

Use a linear equation to solve the word problem.

Evan has \({\$85.00}\) in his piggy bank, and he spends \({\$2.50}\) every day.

Bobbi has \({\$31.00}\) in her piggy bank, and she saves \({\$2.00}\) every day.

If they continue to spend and save money this way, how many days later would they have the same amount of money in their piggy banks?

days later, Evan and Bobbi will have the same amount of money in their piggy banks.

28

Use a linear equation to solve the word problem.

Will has \({\$100.00}\) in his piggy bank, and he spends \({\$4.00}\) every day.

Ross has \({\$34.00}\) in his piggy bank, and he saves \({\$2.00}\) every day.

If they continue to spend and save money this way, how many days later would they have the same amount of money in their piggy banks?

days later, Will and Ross will have the same amount of money in their piggy banks.

29

A hockey team played a total of \(167\) games last season. The number of games they won was \(17\) more than five times of the number of games they lost.

Write and solve an equation to answer the following questions.

The team lost games. The team won games.

30

A hockey team played a total of \(117\) games last season. The number of games they won was \(13\) more than three times of the number of games they lost.

Write and solve an equation to answer the following questions.

The team lost games. The team won games.

31

A rectangleâs perimeter is \({278\ {\rm ft}}\text{.}\) Its length is \({4\ {\rm ft}}\) longer than four times its width. Use an equation to find the rectangleâs length and width.

Its width is .

Its length is .

32

A rectangleâs perimeter is \({226\ {\rm ft}}\text{.}\) Its length is \({1\ {\rm ft}}\) longer than three times its width. Use an equation to find the rectangleâs length and width.

Its width is .

Its length is .

33

Briana has saved \({\$45.00}\) in her piggy bank, and she decided to start spending them. She spends \({\$5.00}\) every \(7\) days. After how many days will she have \({\$30.00}\) left in the piggy bank?

Briana will have \({\$30.00}\) left in her piggy bank after days.

34

Maygen has saved \({\$49.00}\) in her piggy bank, and she decided to start spending them. She spends \({\$5.00}\) every \(6\) days. After how many days will she have \({\$29.00}\) left in the piggy bank?

Maygen will have \({\$29.00}\) left in her piggy bank after days.

35

The following two triangles are similar to each other. Find the length of the missing side.

The length of the side labeled \(x\) is and the length of the side labeled \(y\) is .

37

A restaurant used \({639.4\ {\rm lb}}\) of vegetable oil in \(23\) days. At this rate, \({1306.6\ {\rm lb}}\) of oil will last how many days?

The restaurant will use \({1306.6\ {\rm lb}}\) of vegetable oil in days.

38

A restaurant used \({914.5\ {\rm lb}}\) of vegetable oil in \(31\) days. At this rate, \({1209.5\ {\rm lb}}\) of oil will last how many days?

The restaurant will use \({1209.5\ {\rm lb}}\) of vegetable oil in days.

39

Use a linear equation to solve the word problem.

Massage Heaven and Massage You are competitors. Massage Heaven has \(6500\) registered customers, and it gets approximately \(550\) newly registered customers every month. Massage You has \(8500\) registered customers, and it gets approximately \(450\) newly registered customers every month. How many months would it take Massage Heaven to catch up with Massage You in the number of registered customers?

These two companies would have approximately the same number of registered customers months later.

40

Use a linear equation to solve the word problem.

Two truck rental companies have different rates. V-Haul has a base charge of \({\$70.00}\text{,}\) plus \({\$0.70}\) per mile. W-Haul has a base charge of \({\$60.40}\text{,}\) plus \({\$0.80}\) per mile. For how many miles would these two companies charge the same amount?

If a driver drives miles, those two companies would charge the same amount of money.

41

A rectangleâs perimeter is \({134\ {\rm ft}}\text{.}\) Its length is \({5\ {\rm ft}}\) shorter than three times its width. Use an equation to find the rectangleâs length and width.

Its width is .

Its length is .

42

A rectangleâs perimeter is \({178\ {\rm ft}}\text{.}\) Its length is \({2\ {\rm ft}}\) longer than two times its width. Use an equation to find the rectangleâs length and width.