Systems of Linear Equations

Algebra 2 Tutorial

Systems of Linear Equations


Systems of Linear Equations is a common problem you will see in algebra courses, even up to the college level. You are usually given two linear equations and must find their intersection to solve the system. I find that students don’t often struggle with these problems until they are in the form of a word problem. The main problem I find students struggling with is is identifying variables, and setting up the system.

Sample Problem

Suppose there is a new amusement park with two rides: a roller coaster and a Ferris Wheel. Each ride has its own cost per ticket to board the ride. The roller coaster costs 1.75 dollars per ticket and the Ferris Wheel costs 3.50 dollars per ticket. On a particular day, they sold 700 tickets total, and the park made 1251.25 dollars on that day. A) How many were roller coaster tickets? B) How many were Ferris Wheel tickets? C) Which was the more popular ride that day, and by how much?


First step: Identify the variables and label accordingly.
Let x be the number of roller coaster tickets, and y be the number Ferris Wheel tickets.

Second step: Setup the System
The problem states that in total they sold 700 tickets on that particular day, so we have our first equation. Remember tickets go with tickets.

The problem states the cost per roller coaster ticket was 1.75 dollars and the cost per Ferris Wheel ticket was 3.50 dollars, and that in total they made 1251.25 dollars on that particular day.

Third step: Isolate one variable.
The next step is to isolate one variable, either x or y, it does not really matter which. I will do x on the first equation as it is much simpler than second equation:
x+y=700 (Given)
x+y-y=700-y (Subtraction)
x=700-y (Result)

Fourth step: Substitution
Now that we know what x is, we can substitute it into our other equation to solve for y.
1.75(x)+3.50(y)=1251.25 (Given)
1.75(700-y)+3.50(y)=1251.25 (Substitution)
1225-1.75(y)+3.50(y)=1251.25 (Distributive property)
-1.75(y)+3.50(y)=26.25 (Subtract 1225 from both sides to isolate the y variable)
1.75(y)=26.25 (combine the like terms)
y=15 (divide)

Fifth step: Solve for the other variable.
Now that we have y, we can find the exact value for x.
x+y=700 (Given)
x+15=700 (Substitution)
x=700-15 (Subtract 15 from both sides to isolate x)

Sixth step: Answering the question.
Now we have that x (the number of roller coaster tickets) was 685 and that y (the number of Ferris Wheel tickets) was 15.
A) How many were roller coaster tickets?
Answer: 685
B) How many were Ferris Wheel tickets?
Answer: 15
C) Which was the more popular ride that day, and by how much?
We can see that the roller coaster was more popular as it sold more tickets.
To find out how many more it sold we subtract:
685-15=670. It sold 670 more tickets than the Ferris Wheel.

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