<< Chapter < Page | Chapter >> Page > |
Let’s study some interesting problems that involve linear equations in one variable. In order to solve such problems, we apply the following five-step method:
If the answer doesn’t check, you have either solved the equation incorrectly, or you have developed the wrong equation. Check your method of solution first. If the result does not check, reconsider your equation.
If it has been your experience that word problems are difficult, then follow the five-step method carefully. Most people have difficulty because they neglect step 1.
Always start by INTRODUCING A VARIABLE!
Keep in mind what the variable is representing throughout the problem.
This year an item costs $\$44$ , an increase of $\$3$ over last year’s price. What was last year’s price?
$\begin{array}{l}\begin{array}{lllll}\text{Step}\text{\hspace{0.17em}}1:\hfill & \hfill \text{Let}\text{\hspace{0.17em}}x& =\hfill & \text{last}\text{\hspace{0.17em}}\text{year's}\text{\hspace{0.17em}}\text{price}\text{.}\hfill & \hfill \\ \text{Step}\text{\hspace{0.17em}}2:\hfill & \hfill x+3& =\hfill & 44.\hfill & x+3\text{\hspace{0.17em}}\text{represents}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{\$3}\text{\hspace{0.17em}}\text{increase}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{price}\text{.}\hfill \\ \text{Step}\text{\hspace{0.17em}}3:\hfill & \hfill x+3& =\hfill & 44\hfill & \hfill \\ \hfill & \hfill x+3-3& =\hfill & 44-3\hfill & \hfill \\ \hfill & \hfill x& =\hfill & 41\hfill & \hfill \\ \text{Step}\text{\hspace{0.17em}}4:\hfill & \hfill 41+3& =\hfill & \text{44}\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}\\ \begin{array}{ll}\text{Step}\text{\hspace{0.17em}}5:\hfill & \text{Last}\text{\hspace{0.17em}}\text{year's}\text{\hspace{0.17em}}\text{price}\text{\hspace{0.17em}}\text{was}\text{\hspace{0.17em}}\$41.\hfill \end{array}\end{array}$
This year an item costs $\$23$ , an increase of $\$4$ over last year’s price. What was last year’s price?
Last year's price was $\$19$
The perimeter (length around) of a square is 60 cm (centimeters). Find the length of a side.
$\begin{array}{ll}\text{Step}\text{\hspace{0.17em}}\text{1:}\hfill & \text{Let}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{=}\text{\hspace{0.17em}}\text{length}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{side}\text{.}\hfill \\ \text{Step}\text{\hspace{0.17em}}\text{2:}\hfill & \text{We}\text{\hspace{0.17em}}\text{can}\text{\hspace{0.17em}}\text{draw}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{picture}\text{.}\hfill \end{array}$
$\begin{array}{l}\begin{array}{lllll}\text{Step}\text{\hspace{0.17em}}3:\hfill & \hfill x+x+x+x& =\hfill & 60\hfill & \hfill \\ \hfill & \hfill 4x& =\hfill & 60\hfill & \text{Divide}\text{\hspace{0.17em}}\text{both}\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}4.\hfill \\ \hfill & \hfill x& =\hfill & 15.\hfill & \hfill \\ \text{Step}\text{\hspace{0.17em}}4:\hfill & \hfill 4(15)& =\hfill & 60.\hfill & \hfill \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\end{array}\\ \begin{array}{ll}\text{Step}\text{\hspace{0.17em}}5:\hfill & \text{The}\text{\hspace{0.17em}}\text{length}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{side}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}15\text{\hspace{0.17em}}\text{cm}\text{.}\hfill \end{array}\end{array}$
The perimeter of a triangle is 54 inches. If each side has the same length, find the length of a side.
The length of a side is 18 inches.
Six percent of a number is 54. What is the number?
$\begin{array}{ll}\text{Step}\text{\hspace{0.17em}}1:\hfill & \text{Let}\text{\hspace{0.17em}}x=\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{number}\hfill \\ \text{Step}\text{\hspace{0.17em}}2:\hfill & \text{We}\text{\hspace{0.17em}}\text{must}\text{\hspace{0.17em}}\text{convert}\text{\hspace{0.17em}}6\%\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{decimal.}\hfill \end{array}\text{\hspace{0.17em}}$
$\begin{array}{l}\begin{array}{lllll}\hfill & \hfill 6\%& =\hfill & .06\hfill & \hfill \\ \hfill & \hfill .06x& =\hfill & 54\hfill & .06x\text{\hspace{0.17em}}\text{occurs}\text{\hspace{0.17em}}\text{because}\text{\hspace{0.17em}}\text{we}\text{\hspace{0.17em}}\text{want}\text{\hspace{0.17em}}6\%\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}x\text{.}\hfill \\ \text{Step}\text{\hspace{0.17em}}3:\hfill & \hfill .06x& =\hfill & 54.\hfill & \text{Divide}\text{\hspace{0.17em}}\text{both}\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\mathrm{.06.}\hfill \\ \hfill & \hfill x& =\hfill & \frac{54}{.06}\hfill & \hfill \\ \hfill & \hfill x& =\hfill & 900\hfill & \hfill \\ \text{Step}\text{\hspace{0.17em}}4:\hfill & .06(900)\hfill & =\hfill & 54.\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}\\ \begin{array}{ll}\text{Step}\text{\hspace{0.17em}}5:\hfill & \text{The}\text{\hspace{0.17em}}\text{number}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}900.\hfill \end{array}\end{array}$
Eight percent of a number is 36. What is the number?
The number is 450.
An astronomer notices that one star gives off about $3.6$ times as much energy as another star. Together the stars give off $55.844$ units of energy. How many units of energy does each star emit?
Let $x=$ number of units of energy given off by the less energetic star. Then, $3.6x=$ number of units of energy given off by the more energetic star.
$\begin{array}{l}\begin{array}{lllll}\text{Step}\text{\hspace{0.17em}}\text{2:}\hfill & \hfill x+3.6x& \text{=}\hfill & \mathrm{55.844.}\hfill & \hfill \\ \text{Step}\text{\hspace{0.17em}}\text{3:}\hfill & \hfill x+3.6x& \text{=}\hfill & 55.844\hfill & \hfill \\ \hfill & \hfill 4.6x& \text{=}\hfill & 55.844\hfill & \text{Divide}\text{\hspace{0.17em}}\text{both}\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{4}\text{.6}\text{.}\text{\hspace{0.17em}}\text{A}\text{\hspace{0.17em}}\text{calculator}\text{\hspace{0.17em}}\text{would}\text{\hspace{0.17em}}\text{be}\text{\hspace{0.17em}}\text{useful}\text{\hspace{0.17em}}\text{at}\text{\hspace{0.17em}}\text{this}\hfill \\ \hfill & \hfill & \hfill & \hfill & \text{point}\text{.}\hfill \\ \hfill & \hfill x& \text{=}\hfill & \frac{55.844}{4.6}\hfill & \hfill \\ \hfill & \hfill x& \text{=}\hfill & 12.14\hfill & \text{The}\text{\hspace{0.17em}}\text{wording}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{problem}\text{\hspace{0.17em}}\text{implies}\text{\hspace{0.17em}}two\text{\hspace{0.17em}}\text{numbers}\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{needed}\hfill \\ \hfill & \hfill & \text{=}\hfill & \hfill & \text{for}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{complete}\text{\hspace{0.17em}}\text{solution}\text{.}\text{\hspace{0.17em}}\text{We}\text{\hspace{0.17em}}\text{need}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{number}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{units}\text{\hspace{0.17em}}\text{of}\hfill \\ \hfill & \hfill & \hfill & \hfill & \text{energy}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{other}\text{\hspace{0.17em}}\text{star.}\hfill \\ \hfill & \hfill 3.6x& \text{=}\hfill & 3.6(12.14)\hfill & \hfill \\ \hfill & \hfill & \text{=}\hfill & 43.704\hfill & \hfill \\ \text{Step}\text{\hspace{0.17em}}\text{4:}\hfill & \hfill 12.14+43.704& \text{=}\hfill & \mathrm{55.844.}\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\text{\hspace{0.17em}}\hfill \end{array}\\ \begin{array}{ll}\text{Step}\text{\hspace{0.17em}}5:\hfill & \text{One}\text{\hspace{0.17em}}\text{star}\text{\hspace{0.17em}}\text{gives}\text{\hspace{0.17em}}\text{off}\text{\hspace{0.17em}}12.14\text{\hspace{0.17em}}\text{units}\text{}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{energy}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{other}\text{\hspace{0.17em}}\text{star}\text{\hspace{0.17em}}\text{gives}\text{\hspace{0.17em}}\text{off}\text{\hspace{0.17em}}43.704\text{\hspace{0.17em}}\text{units}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{energy}\text{.}\hfill \end{array}\end{array}$
Notification Switch
Would you like to follow the 'Elementary algebra' conversation and receive update notifications?