Complete Guide: Systems of Equations on the ACT® Math
Read time: 4 minutes 30 seconds Last updated: September 23rd, 2024
Systems of equations show up consistently, maybe once or twice per test, in the 20-50 range difficulty.
They're actually some of the more intuitive problems on the ACT Math®. They don't typically require too many steps or much complicated math at all. But this sort of system of equation logic isn't taught frequently in high schools, from my experience. The question requires you to know one new thing, which is how to deal with so-called "systems of equations."
If you have two equations with the same variable, you can stack them. Then, if is a real number, you have "real solutions." This makes more sense when you look at some examples.
Consider This Example
1x + 2 = 3
2x + 1 = 3
Here, if the same value of x satisfies both equations, then you have a real solution. It's as simple as that.
1x + 2 = 3
-2 = -2
x = 1
Let's check to make sure that's right by plugging
into the second equation
2x + 1 = 3
(2 · 1) + 1 = 3
2 + 1 = 3
3 = 3
So, x=1 in both equations. Therefore, the system of equations has a real solution, 1.
Maybe you got a different value for x. That's totally fine. Not only does the system of equations above have a real solution, it has an infinite number of real solutions. So any of value of x will be the same for each equation.
More Practice
Example 1:
Find real solutions, if any exist, for the following equations: 3x + 4 = 10, 6x - 2 = 16
Click for the Answer
3x + 4 = 10
3x = 6
x = 2
6(2) - 2 = 16
12 - 2 = 16
10 = 16
No real solution here, as the equations do not match. The equations don't match because they lead to different values of x.
Example 2:
Find real solutions, if any exist, for the following equations: 5x + 3 = 18, 10x + 6 = 36
Click for the Answer
5x + 3 = 18
5x = 15
x = 3
10(3) + 6 = 36
30 + 6 = 36
36 = 36
Here, for both equations, so you have a real solution.
How will this show up on the ACT® Math?
The ACT Math® Section will typically make the questions as simple as above. They will ask whether or not there is a real solution, and less frequently what that real solution is.
The ACT Math® section hasn't asked about systems of equations with graphs. They might ask you to think about systems of equations as if they were graphs, even without providing actual graphs. I'll elaborate.
If we think about and as lines, they intersect once, when . That's why there's one real solution, 1.
If there are no real solutions, like in lines and , these lines are parallel. They never actually touch. That's why never equals . This is why we say there are no real solutions when the lines are parallel.
But what if the same value of x always satisfies both equations, no matter what that value is? That would mean that for a given system of equations, there is an infinite number of solutions. and , for example, would always be the same exact line for every value of because they're the same exact line.
and
Conclusion
The ACT® Math Section will occasionally expect you to determine whether there are 1 real solution, no real solutions, or an infinite number of solutions.
Keep in mind the logic above regarding how to understand each as either being the same in one case, no cases, or every case respectively. Remember, you can always use the provided scratch paper to sketch out the lines if it helps you visualize the problem.