Baby Algebra Upgrade: How Many Solutions Does A System Of Linear Equations Have?

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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 “easy” 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.) This week we’re building on our new ability to recognize systems of linear equations hiding in seemingly harmless word problems.

Now that you understand that more scenarios than you originally thought are actually systems of linear equations (and since you now understand linear equations), here’s one of the biggies: how many solutions can a system of linear equations have?

This one matters (even if you’re not a math, uh, enthusiast like I am) because it’s a fast and easy way to narrow down your answer choices on the SAT. In fact, you might not even need to actually find the solutions if you can figure out a few things about them right off the bat.

So, think of it this way: if you were holding up two pencils (two lines), there are really only a few options for how they interact.

1) The pencils will eventually cross somewhere, at a specific point.

2) The pencils will never cross, no matter how far you extend them in their straight paths.

3) The pencils are one on top of the other, and therefore completely overlap!

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This is really what we’re talking about when we say “number of solutions.”

  1. Our first scenario had 1 single solution (the point where the pencils cross).

  2. Our second scenario had 0 solutions (the pencils are parallel, so they’ll NEVER meet).

  3. Our third scenario had infinite solutions (every point on one pencil overlapped with every point on the other).

Let’s break this down further.

  1. When two lines cross (“have 1 solution”), that means they must have different slopes.

    (Their y-intercepts totally don’t matter.)

  2. When two lines are parallel (“have no solutions”), that means they MUST have the SAME slopes yet DIFFERENT y-intercepts.

  3. When two lines have “infinite solutions”...it means they’re the SAME LINE, i.e. they have the SAME slope and SAME y-intercepts.

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Now, here’s the crazy shortcut I use:

When I write my linear equations in Standard Form, like this:

Ax + By = C

and

Dx + Ey = F

the slopes are determined by the coefficients of the variables (A, B, D and E), and the y-intercepts are determined by B, E, C and F.

So here’s the biggest trick: the slopes are the same if A and D are proportional to B and E!

In other words, if I have these…

2x + 5y = 9

4x + 10y = 15

…I can make the proportion 2/4 = 5/10, reduce both sides to realize they both equal 1⁄2 and are therefore the same, and realize these two equations have the same slope!

Now, if they have the same slope, they’re either parallel and have NO solution, or they’re the same and have INFINITE solutions.

How can I tell which one it is? Simple! See if B and E are also proportional to C and F! Let’s check…

5/9 = 10/15

...NOPE! Not equal! So their y-intercepts are different. Which means they are parallel. Which means there are NO solutions to this system.

See? Really not that tough. And being able to see how many solutions you’re looking for in a system of linear equations will win you points and save you time on the SAT. Score!

So now, you understand all the tricky ways you could be tested on lines and systems of linear equations. You know all the ways a line could be represented. You know what the numbers in the equations actually MEAN in real-life scenarios. You know which situations give you a SYSTEM of linear equations, and you also know how many solutions that system will even have! That’s actually QUITE a LOT! Especially on the SAT!

But there are a couple of “Baby Algebra” topics that my students tend to brush off because they seem unimportant—and then they end up missing it! So stay tuned, and next week I’ll entirely upgrade your understanding of Algebra’s neglected stepchildren: Inequalities and Absolute Value!