Baby Algebra Upgrade: How To Recognize Systems of Linear Equations

Welcome back to the Ivy Lounge Test Prep series on upgrading your baby Algebra skills! In this series, I’m helping you level up the math skills that are supposed to be “simple” so that you can pick up the (many) points that the SAT and ACT offer you for applying these apparently simple skills in sophisticated ways. (Check out this refresher course on why you may not know the math you think you know, if you missed the beginning of the series.)

Now that you can’t get broadsided by lines and linear equations, guess what the next logical step of upgrading your “baby Algebra” is?

If you said, “putting more than one line together,” you’d be correct!

Let’s review and upgrade your knowledge of one of my favorite topics ever: systems of linear equations!

Baby Algebra Upgrade #3: What ARE Systems of Linear Equations?

OK, so you probably think of these as that “special” type of math problem that you did when you were first learning Algebra, and never again. After all, why do I care where two lines meet? Like, what’s the point? (Pun intended!)

There, there. If you read my last post about understanding that almost every simple equation you come across is actually a line, then you may then see the logic: if you have TWO simple equations, only involving the SAME TWO variables, THAT, my friend, is a system of linear equations!

Here are hidden systems of linear equations:

2x + 3y = 5   & y = 3x – 6

(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. No, they didn’t have to be “x” and “y.”)

Make sense?

Now, here are some SCENARIOS that are really systems of linear equations:

Dan buys three veggie burgers and two orders of sweet potato fries and pays \$18. Sarah buys two veggie burgers and six orders of sweet potato fries and pays \$26. How much is one veggie burger?

or…

Chris can write one song and record two songs in 5 hours. Chris 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 him to write one song and record it?

So, those don’t look like systems of linear equation problems, right? They look like word problems! But if you actually translate these sentences to equations, we find that they’re systems of linear equations in disguise.

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When you translate those scenarios back into equations, you can see what I mean.

Sarah and Dan’s night out becomes:

3b + 2f = \$18

2b + 6f + \$26

(where “b” = “burgers” and “f” = “fries”)

Chris’s musical pursuits become:

w + 2r = 5 hours

3w + 3r = 9 hours

where “w” = “songs written” and “r” = “songs recorded”

See it now? Each one has: TWO different equations using the SAME TWO VARIABLES! It’s a system of linear equations, pure and simple! Once you realize this, you can use any of the methods you learned in Algebra to solve it: substitution, graphing, or elimination!

This is a simple conceptual upgrade that makes a big difference in your ability to understand and solve the system.

By realizing 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 my next post.