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 show up 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!)
The previous post in this series taught you that a lot of different scenarios are actually systems of linear equations in disguise (and since you now know what linear equations are to begin with), on to the next Q you need to master: how many solutions can a system of linear equations have?
This one is worth learning (even if you’re not a math mega-fan like moi) because it’s a quick and easy way to eliminate 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!
This is actually what we’re referring to when we ask about the “number of solutions” that a given system of linear equations has.
Our first scenario had 1 single solution (the point where the pencils cross).
Our second scenario had 0 solutions (the pencils are parallel, so they’ll NEVER meet).
Our third scenario had infinite solutions (every point on one pencil overlapped with every point on the other).
Let’s break this down further.
When two lines cross (“have 1 solution”), that means they must have different slopes.
(Their y-intercepts totally don’t matter.)
When two lines are parallel (“have no solutions”), that means they MUST have the SAME slopes yet DIFFERENT y-intercepts.
When two lines have “infinite solutions”...it means they’re the SAME LINE, i.e. they have the SAME slope and SAME y-intercepts.
Now, here’s the secret (well, no longer a secret to YOU!) shortcut I use:
If you write a linear equation 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 deduce which one is the case? 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.
Not so tricky after all, right? And being able to figure out how many solutions you’re looking for in a system of linear equations will earn you points and save you time on the SAT. Yesss!
So now, you understand all the thorny 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 in two weeks I’ll entirely upgrade your understanding of Algebra’s neglected stepchildren: Inequalities and Absolute Values!
BTW, if you want to see your Math (and other SAT/ACT section) scores soar, working with me one-on-one is the most effective option there is. Alternately, if you prefer to study on your own schedule, check out my SAT Math Cram Plan ebook (with an ACT version here).
