System of Equations

System of Equations involves taking a series (usually two) of equations and finding their “solution”, meaning variable values that make all equations in the series true. Another way to think of this is to imagine graphing those equations, and a system of equations problems asking students to find the intersection points between the given equations.

Two Methods for Systems of Equations: Elimination and Substitution

Substitution

The substitution method of system of equations involves isolating a variable in one of the equations, and then plugging in that expression for the variable into the other equation.

For example, if a question gives the following system of equations:

2x + 3y  = 2

x − 2y = 8

Then to substitute, a student must choose one equation and one variable to isolate. Isolating the x variable in the bottom equation gives x  = 2y + 8. That expression is then substituted in for the x variable in the top equation: 2(2y+8) + 3y = 2. This then leads to 7y=  − 14 or y =  − 2.

Elimination

Just like with any algebraic equation of coure, one can either work with variables individually or with the equation as a whole. With systems of equations, questions are generally looking for solutions that make both equations true. Since that means the equations must be true statements, the two equations can be multiplied by, divided by, subtracted from, or added to each other, just like any other true statements.

The reason the word “elimination” is used to refer to this is because the most common way to make the two equations in a system of equations interact directly is to subtract one equation from another. Specifically, a student should look to subtract one equation from another such that one of the two variables in those equations is eliminated.

To do this, the variable should have the same coefficient but opposite signs in each equation. This may already be true when the equations are first stated in the problem, or one equation may need to be manipulated to make this work.

In the example above, neither variable has the same coefficient but opposite signs, so a little work would need to be done.

The bottom equation could be multiplied by -2, which would lead to the x variables having same coefficients and opposite signs. A student could also look to eliminate the y variable, but this would require either multiplying one equation by a non-whole number or manipulating both equations. With the ACT, speed is always of the essence, so students should try to choose the quickest/most efficient path!

Therefore, the bottom equation is multiplied by -2:  − 2x + 4y =  − 16

The two equations are then added together. Technically either side can be added together, but it’s easiest to add the variable sides and constant sides together.

7y = -16 is our sum, or y =  − 2. Same answer as our previous method, but just done a little differently.

Deciding Between Substitution and Elimination

Students should do what they’re most comfortable and confident with.

That being said, there are some general guidelines for choosing between these two methods, if a student is comfortable with either approach.

Look at the Numbers in the Equation

First, look at the numbers in the equations. As the following questions:

• Are there any shared coefficients and opposite signs in front of the variables?

If yes, that means no work is needed and elimination will be the easiest route.

• Are the constants and coefficients factors of each other? Would they divide into each other evenly or cleanly?

If yes, then it could be easy to manipulate the equation and utilize elimination. 

If no, then it could be difficult to eliminate numbers easily. substitution may be best.

Look at the Question

What is the question asking for?

• Does it want the value of a single variable or of multiple variables or the product/sum of two variables?

If it is asking for a single variable, which cannot be easily reached through elimination, then substitution is often helpful.

If it is asking for the product or sum of multiple variables,  especially if that product or sum can be reached through elimination or by leaving an equation intact.

Word Problems

Sometimes problems will involve systems of equations, but the equations will not be directly given by the question. Instead, the problem might be given as a word problem, but where the information presented would clearly lead to the creation of a pair of equations. With these equations there should be two variables, which can be used to calculate two different things: time and distance, for example.

Here is an example question from the April 2021 ACT.

20 = A + B where A and B are the number of courses Susan received an A or B in, respectively

3.15 = 4A +3B20. Again, we can use either substitution or elimination here.

 − 3(20 = A + B) ⇒  − 60 =  − 3A − 3B

20 (

3.15 = 4A +3B20) ⇒ 63 = 4A + 3B

3 = A Susan received 3 A’s

One important note: since the question asked for the value of A, B was the variable that the student cancelled out. If A had been cancelled out, then that would have added an extra (and unnecessary) step.

Cancel out or substitute in for the variable that you’re not solving for: this will leave just the variable you’re solving for in the equation.