We can analyze the world around us by modeling situations using equations. Modeling means to write down an equation that fits the scenario, and then solve it to find some quantity that is of interest. We can use linear equations to model specific situations.
To solve a modeling problem, we should:
Read the problem from start to finish.
Look for a question within the problem (usually at the end).
Start by writing, "Let something." Usually, is some unknown quantity that the question is asking for.
Reread the question, and try to write down an equation for that must be true.
Solve the equation.
Jim has in his pocket, which is two-thirds of the price of a particular book. How much does the book cost?
First, let the cost of the book, which is the unknown quantity we're trying to find.
We note that:
Jim has in his pocket.
Two-thirds of the cost of the book corresponds to
Since two-thirds of the cost of the book is equal to the amount of money that Jim has, the equation for is
We can solve this equation using the multiplication principle:
Therefore, the book costs
Jack bought a pen for $3$ dollars along with four mechanical pencils. He spent a total of $11$ dollars. How much did each pencil cost?
|
a
|
$1$ dollars |
|
b
|
$50$ cents |
|
c
|
$2$ dollars |
|
d
|
$4$ dollars |
|
e
|
$3$ dollars |
Triple a number minus $4$ is equal to $17.$ Find the number.
|
a
|
$4$ |
|
b
|
$5$ |
|
c
|
$9$ |
|
d
|
$8$ |
|
e
|
$7$ |
A math test has questions, and each correct answer is worth points. To calculate the final grade, the teacher first subtracts the number of incorrect answers from the total points earned from correct answers and then divides the result by If Ron gets a final grade of how many questions did he answer correctly?
Let the number of questions Ron answered correctly, which is the unknown quantity we're trying to find.
We note that:
The total number of points obtained from the correct answers is
The number of incorrect answers is
The teacher subtracts the number of incorrect answers from the total points obtained from the correct answers, which gives
Then, the teacher divides the number obtained above by So, the final grade is
Ron's final grade is So, we have the equation
We can solve this equation using the addition and multiplication principles:
Thus, Ron answered questions correctly.
On Tuesday, Haley bought ten pencils to add to her collection. On Wednesday, she lost half of the pencils, leaving her with only fifteen. How many did she have on Monday?
|
a
|
$25$ |
|
b
|
$10$ |
|
c
|
$15$ |
|
d
|
$20$ |
|
e
|
$30$ |
If we subtract $14$ from a certain number and take one-quarter of the result, we obtain twice the number. Find the number.
|
a
|
$3$ |
|
b
|
$-2$ |
|
c
|
$7$ |
|
d
|
$-7$ |
|
e
|
$2$ |
Triple a number is greater than one-third of that number. Find the number.
Let the number we are being asked to find.
We note that:
Triple the number corresponds to
One-third of the number corresponds to
Now, we link both expressions using the fact that the triple the number is greater than one-third of the number, which corresponds to the equation
We solve the equation using the addition and multiplication principles:
Therefore, the number we are interested in is
Find a number whose half subtracted from its triple is equal to $25.$
|
a
|
$10$ |
|
b
|
$12$ |
|
c
|
$17$ |
|
d
|
$11$ |
|
e
|
$8$ |
Find a number whose triple subtracted from $42$ is equal to its double plus $12.$
|
a
|
$6$ |
|
b
|
$5$ |
|
c
|
$2$ |
|
d
|
$4$ |
|
e
|
$7$ |