Solving a word problem using a system of linear equations of the form Ax + By = C

Question

One month Elsa rented 2 movies and 3 video games for a total of $26. The next month she rented 6 movies and 5 video games for a total of $56. Find the rental cost for each movie and each video game.

Answer 1

With this problem our goal is to get two equations in the form of Ax + By = C where x is the price of renting a movie and y the price of renting a video game. A, B, and C are simply constants, or numbers that won't change in value. Once we have two equations, we can use them to solve for x and y.

1) Write the equation for Elsa's first month

If x is the price of renting a movie and Elsa rents two movies, her movie rental cost comes out to 2x (number of movies multiplied by the price of each movie). Similarly, for video games costing y dollars to rent, her cost is 3y. Her total is $26 which means her movie cost (2x) and her video game cost (3y) together come to $26. This gives us 2x + 3y = $26

With just one equation and two variables, we can't solve for either variable. We need a second equation.

2) Write the equation for Elsa's second month

Following the same steps we took to write the first equation, we get 6x + 5y = $56

Now we have two equations with the same two variables, we can use those equations to solve both variables.

3) Using either equation, solve for one of the variables

Now, we want to isolate a variable. Looking at the first equation, let's solve for x. Solving for x gives us: x = (26 - 3y) / 2

4) Solve for the other variable by substitution

Now that we have a value for x in terms of y, we can substitute that in for x in the second equation to give us one equation with one variable. It should look like: 6 (26 - 3y)/2 + 5y = 56. With some algebra, we simplify down to 78 -9y + 5y = 56 and further to -4y = -22. That's an easy solve for y. Divide -22 by -4 to get $5.50 for the cost of renting a video game.

5) Substitute the solved variable's value into the equation

Almost done! Plug in the value of y into either equation and then solve for x. Returning to the first equation, we get 2x + 3($5.50) = $26 or, 2x + $16.50 = $26. Now with one variable and one equation x = $______

(x = $4.75)

6) Check your work.

Plug both values into the other equation and make sure they make sense. Taking the second equation, we get 6($4.75) + 5($5.50) = $56. Awesome!