Whenever the word "fractions" comes up in math classes, there are always students who shudder and dismay. Now, the notion of combining fractional equations with systems of equations might seem especially terrifying. But I hope to convince you that it's really not so bad...honestly! We'll take what we already know about fractional expressions and fractional equations, and apply it to systems of equations. Let's start with a simple example.

Example #1

+ = 4
+ y = 4

What do you do if you have a fractional equation? You find the least common denominator, and multiply each equation by that denominator. The LCD of the first equation is 6, so we'll do this:

6( + ) = 6(4)
3x + 2y = 24

In the second equation, the LCD is also 6, so we'll multiply both sides by 6:

6( + y) = 6(4)
x + 6y = 24

Now you have two equations without fractions, and you can finish solving.

Example #2

+ = 11
- = 9

Solution #2

Sometimes you don't have to find LCDs in order to clear the fractions. Notice that in this example, the coefficients of x are and -, which means that if we add the two equations, the y variables will cancel, and since + = x, we have:

x = 11 + 9 = 20. Then we can finish by solving for y.

Example #3

+ = 6
- = 3

Solution #3

This problem is different because there are variables in the denominators. In this case, clearing the denominators by multiplying by the LCD will result in introducing an xy term on the right, which might get messy, so let's see if we can use the previous method. We'll try to use addition method to get rid of a variable:

+ = 9
= 9
= 9
x = 2

Now we can plug that back into an equation:

+ = 6
+ = 6
5 + = 6
= 1
y = 3

1.

Finish solving example #1.

2.

Finish solving example #2.

3.

Solve the system: + = 9; + = 4

4.

Solve the system: + = 5; - = 3

5.

Solve the system: + = ; + y =

6.

Solve the system: + = 3; - = 4

7.

Solve the system: + = 10; + = 5