Hello, and welcome back to the Ivy Lounge Test Prep™ series on refreshing your Baby Algebra! These posts will teach you to spruce up the math skills that you supposedly learned in Middle School—and which have suddenly become relevant to your life again as they’re tested on the SAT/ACT. The goal is to help you grab every last one of the (many) points that the SAT and ACT offer those who know how to apply these so-called basic skills in higher-level ways. (If today’s post is the first you’re hearing of this series, I’d recommend you first consult this starter post on why you may not know the math you think you know!)
Since the last post in this series reminded you how to tackle lines and linear equations, the next logical step of updating your “baby Algebra” is….
That’s right! Putting more than one line together.
So, come with me as we review and update your knowledge of a subject that I, personally, find to be really fun: systems of linear equations!
What ARE Systems of Linear Equations?
There’s a good chance you think of these as a very specific genre of math problem that you practiced in your earliest days of Algebra….and then never touched again. After all, why does it matter where two lines meet? I mean, what’s the point? (Pun intended!)
But here’s a past post of mine that might help clear up why this matters: nearly every simple equation you encounter is actually a line. So if you have TWO simple equations, only involving the SAME TWO variables—well, THAT, my friend, is a system of linear equations!
Here are hidden systems of linear equations:
4x + y = 5 & y = 4x – 5
(Two different equations, using the same two variables: x and y. It doesn’t matter that the first is written in Standard Form and the second is written in Slope-Intercept Form.)
k = 7 & n = ½ k + 7
(Two different equations, using only “k” and “n”…two different variables. No, I did not have to use BOTH variables in each equation. And no, they didn’t have to be “x” and “y.”)
With me so far?
Now, here are some SCENARIOS that are really systems of linear equations:
Eddie buys three burritos and two orders of onion rings and pays $18. Mira buys two burritos and six orders of onion rings and pays $26. How much is one burrito?
or…
Miguel can write one song on his guitar and record two songs in 5 hours. He can write three songs and record three songs in 9 hours. Assuming it always takes him the same amount of time to write a song or to record a song, how long would it take Miguel to write one song and record it?
At first glance, these two paragraphs don’t seem like systems of linear equation problems, right? They just look like word problems! But if you actually write these sentences out as equations, we find that they’re undercover systems of linear equations.
When you translate those scenarios back into equations, you can see what I mean.
Eddie and Mira’s tasty meal becomes:
3b + 2o = $18
2b + 6o = $26
(where “b” = “burritos” and “o” = “onion rings”)
Miguel’s guitar hobby becomes:
w + 2r = 5 hours
3w + 3r = 9 hours
where “w” = “songs written” and “r” = “songs recorded”
Notice anything? Each scenario has TWO different equations, both of which use the SAME TWO VARIABLES! In other words—it’s a system of linear equations! Once you grasp this, you can use any of the methods you learned in Algebra to solve it: graphing, elimination, or substitution!
This is a simple conceptual level-up that massively improves your ability to understand and solve the system.
Once you realize that most word problems and “easy” equations are really systems of linear equations in disguise, not only can you solve them—but you will know HOW MANY solutions you’re even trying to find! That’s exactly what we’ll talk about in the next post of this series.
And if you’d like SAT or ACT help with more than just Baby Algebra—help catered to your specific problem areas!—consider working with me one-on-one. Or, if you prefer to fly solo, check out my one-stop SAT Math Cram Plan ebook (or the ACT version here).
