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.
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}???
Let’s look at a few more.
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}???
One last example.
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}???