Systems of Equations Part 1: What are Systems of Equations?

Clouds in the sky. Systems of equations can be used to forecast the weather.

Image by Matthew Henry

How do meteorologists forecast the weather? They use systems of equations.

Sometimes you encounter a problem that requires more than one related equation in order to solve for all of the unknowns. These equations form a system, meaning that the equations all share the same variables. Not every equation in a system has to have all of the variables — there just has to be enough overlap for you to solve.

One common system of equations that you will run into includes one equation for the total number of items you have and another equation for the total price you paid for all of the items. Most of the systems that you will run into on the ACT® will consist of linear equations, which have a highest exponent of 1 and graph as a line (ex: y = 2x + 5). You might also see systems that include one or more quadratic equations, which have a highest exponent of 2 and graph as a U-shaped parabola (\( y = 3x^2 + 6 \)). These might not be all of the kinds of equations you’ll see in a system, but they are the most common.

Ways to solve

If you’re trying to solve a problem with more than one unknown, the rule of thumb is that you need as many different equations as you have unknowns in order to solve for everything. For example, if you don’t know the values of x and y, you would need two equations to solve for both x and y. 

One of the interesting things about systems of equations is that there are lots of ways to solve them. Here are four approaches that you can use:

Graphing: A graphing calculator can help you visualize a system of equations. You can also use Desmos to graph equations for free. If a problem asks you to find whether two or more equations intersect, plugging them into a graphing calculator often can be easier and faster than going through a lot of algebra by hand. This strategy can also be useful if your numbers are difficult to manipulate.

For example, if you are asked whether \(5x – 3y = 7\) and \( x^2 + 3 \) intersect, because the equations are two different types (one linear equation, one quadratic equation), it is probably fastest to graph that system.

A graph of a system of equations.

From this graph, it is easy to see that the two equations don’t intersect (there is no real solution to the system).

You can use this same strategy if you are given a real-world scenario. You can assign x and y to values you don’t know, then find out what they are by looking for the intersection point of the equations in your system on a graph.

For example, let’s say your friend went online stress shopping before finals and bought 7 pairs of flip flops. All of the flip flops are either yellow or green (your friend’s favorite colors). Each pair of yellow flip flops cost $18.00, and each pair of green flip flops cost $21.00 (including shipping). If you friend spent $141, how many green flip flops did they buy?

You can solve this a number of ways, but one approach is to say that yellow flip flops = y and green flip flops = x (or the other way around — you just have to stay consistent across equations). Then, write a system of equations and graph it:

x green + y yellow = 7 total pairs

$21x + $18y = $141

If you graph x + y = 7 and 21x + 18y = 141, you can see that the intersection point of the two lines is (5, 2).

Learn how to solve systems of equations using graphs.

This means that your friend bought 5 green pairs of flip flops, because you wrote your equations with the x value standing for green flip flops.

Setting Equal: If it is easier to solve a system by hand, then setting two equations equal to each other is one way to find intersection point(s). Use this strategy if a question says that two equations should have equal value.

For example, let’s say you are comparing memberships for the gym. One is $15 per month with a $30 sign up fee, and the other is $13 per month with a $40 sign up fee. If you want to know when the memberships will cost the same amount, write two equations, then set them equal to each other. The two membership plans’ costs can be expressed with the following expressions, where n is the number of months from when you signed up.

Plan 1: 15*n + 30

Plan 2: 13*n + 40

Set these two expressions equal to each other.

15n + 30 = 13n + 40

Combine like terms.

2n = 10

Solve for n.

n = 5 months.

Substitution: A common method of solving a system of equations is to solve for a variable in terms of the others in one equation and then substitute this expression into the other equation. This method is most useful when one of your equations is easy to manipulate.

For example, let’s say you know that x – y = 9 and you want to find out what x + y equals if        12x – 10y – 2 = 100.

Manipulate the first equation to get x in terms of y.

x = 9 + y

Substitute this expression in for x in the second equation.

12(9 + y) – 10y – 2 = 150

108 + 12y – 10y  = 152

2y = 44

y = 22

Now, plug y = 22 into the first equation and solve for x.

x – 22 = 9

x = 31

Finally, we can add x and y to get our answer.

31 + 22 = 53

Elimination: Elimination involves adding two equations together and knocking out an unknown in the process. This is a useful approach if it is easy to make a variable in two equations have the same coefficient with opposite signs (like 3x and -3x). This only works with linear equations, so don’t use this strategy for quadratic equations.

For example, find the values of x and y in the following system:

3x + 6y = 45

-x + 4y = 9

If we multiply every value in the second equation by 3, we will knock out the x’s if we add the two equations together.

  3x + 6y = 45

-3x + 12y = 27

          18y = 72

             y = 4

We can then plug 4 in for y in either equation to find that x = 7.