This problem is a system of linear equations word problem. That means you have two variables that will appear in two linear equations, which will intersect in only one point to give you your answer to each variable. The hardest part of these problems is to recognize the variables and equations to set up.
For all word problems, we start the same way. We make a list of what is given and write down what we are asked to find.
Given:
Car 1 has an efficiency of 35 mi/gal.
Car 2 has an efficiency of 15 mi/gal.
Both cars went a total of 1125 mi in one week.
Both cars used a total of 55 gallons in one week.
We are looking for: How many gallons of gas does each car use during the week?
Let's call the gallons of gas used in Car 1 x, and the gallons of gas used in Car 2 y. Now we have two variables. We need to put both variables into two linear equations to make our system of equations. How can we do that?
The easier of the two equations to identify (in my opinion) is based on the given information that both cars used a total of 55 gal in one week. A total just means to add two items. Since 55 gal were used by both cars in one week, and we are looking for x gal of gas and y gallons of gas, we can write:
x + y = 55
Hopefully, that was pretty straightforward. Now for the tougher one.
We were also told that 1125 mi were driven by both cars over the week. Since mi does not equal gal, we need something to make a conversion with. What else were we given? Oh, yes! We were given mi/gal efficiencies for each car! If we multiply mi/gal by gal used for each car, that gives us mi. mi plus mi = mi, so we can write that last equation.
Remember that the units of both x and y are gallons. So, for Car 1, (35 mi/gal) * x gal will allow the gal to cancel out, leaving us with 35x miles. The same thing will happen for Car 2, (15 mi/gal) * y gal will allow the gal to cancel out, leaving us with 15y miles. Since the total miles given was 1125 mi, we just add the two terms we just created together to make 1125. Therefore, our second equation is:
35x + 15y = 1125
If you were able to follow that reasoning, you now have two equations with two unknowns to solve the problem with. If you have questions, please ask because this setup is the hardest part of the problem.
Now, we write the two equations together, like this:
x + y = 55
35x + 15y = 1125
There are two ways to solve a system like this. You can use the Substitution Method or the Elimination Method. The Elimination Method would require you to multiply the top equation (both sides) by either -35 to eliminate x or by -15 to eliminate y from the picture. Then, you would add the two equations together, solve for the one remaining variable, and replace that value in for either x or y in the bottom equation to solve for the other variable.
I preferred to use the Substitution Method for this one because of the simple first equation, x+y = 55. The first step would be to solve this equation for either x or y. Either is fine, I chose to solve for y.
y = 55 - x
Next, because this is called substitution, we substitute 55 - x in for y in the other original equation. This would look like:
35x + 15(55-x) = 1125.
Distribute the 15 through the 55-x. That's a large enough number to use a calculator for. You should get:
35x + 825 - 15x = 1125.
Combine the x-terms to get:
20x + 825 = 1125.
Subtract 825 from both sides to get:
20x = 300.
Divide both sides by 20 to get:
x = 15
Finally, we need to substitute 15 in for x in one of the original equations to solve for y. The first equation is easier:
15 + y = 55.
Subtract 15 from both sides:
y = 40
Since x referred to Car 1 and y referred to Car 2, Car 1 used 15 gal and Car 2 used 40 gal over the week.
