How to solve equation modeling Algebra problems

What is equation modeling?

In this lesson we’ll look at how to take a description of an equation in words and transform it into a written equation by using a table.

We’ll also look at how to combine existing equations with new information to better model the situation.

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How to turn problems into equations with equation modeling

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Solving equation modeling problems in Algebra

Example

An RV and a motorcycle were driven for a month. The motorcycle traveled $1,000$ miles more than the RV. The fuel mileage for the RV was $15$ miles per gallon (mpg), and the fuel mileage for the motorcycle was $43$ mpg.

Write an equation which gives the total amount of fuel, $g$ (in gallons), that was used by the two vehicles during that month in terms of the distance $m$ (in miles) traveled by the motorcycle. We can use a table to show the fuel mileages and distances traveled.

We’ll start by writing an equation that gives the distance $r$ (in miles) traveled by the RV in terms of $m$. We know that $m=r+1,000$ because the motorcycle traveled $1,000$ miles more than the RV. So $r=m-1,000$, and we’ll replace “$r$ miles” in our table with “$m-1,000$ miles.”

You can calculate the gallons used by dividing the distance by the mileage. So we get

Now we know that 

$g= \frac{m-1,000}{15}+\frac{m}{43}$

$g=\frac{43}{43}\cdot \frac{m-1,000}{15}+\frac{15}{15} \cdot \frac{m}{43}$

$g= \frac{43m-43,000}{645}+\frac{15m}{645}$

$g= \frac{43m-43,000+15m}{645}$

$g=\frac{58m-43,000}{645}$

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Let’s look at a few more.

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we’ll look at how to take a description of an equation in words and transform it into a written equation by using a table.

Example

A rock is thrown at a speed of $16$ ft/s straight downward from a high platform. The distance it travels can be calculated using $D=16t^2+16t$, where $t$ is the amount of time in seconds that it’s been falling. The average speed of any object can be calculated using $V=D/t$. Write an equation giving the time of fall in terms of $V$.

Start with $v=D/t$ and substitute $16t^2+12t$ for $D$.

$V=\frac{16t^2+16t}{t}$

$V=16t+16$

Solve for $t$ in terms of $V$.

$16t=V-16$

$t=\frac{V-16}{16}$

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One last example.

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Example

Each employee at a certain level of employment at a company is paid $42,000.00$. The owner of the company wants to divide $120,000.00$ evenly among these employees over the course of a year. Write an expression in terms of the number of employees $e$, that gives the amount $a$ each employee earns per month.

Each employee earns a monthly salary of $42,000.00 \div 12 = 3,500.00$. So each employee earns $3,500.00$ per month.

Now the bonus is $120,000.00 \div 12 = 10,000.00$ each month, but it’s divided by the number of employees $e$. So the monthly amount of the bonus for each employee is

$\frac{10,000.00}{e}$

The total amount each employee earns per month is then

$a=3,500 + \frac{10,000.00}{e}$

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