Baby Algebra Upgrade: How To Recognize (And Actually Understand) A Linear Equation

Welcome back to the Ivy Lounge Test Prep series on the SAT “math paradox” and the math upgrades you need to beat it! 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 offers you for applying these “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—it explains what it is we’re doing here, and why it will help you snag math points you might otherwise miss!)

So now that you know that we’re here to upgrade your baby Algebra so you can use it in the sophisticated ways the SAT (and, to a lesser extent, the ACT) demands…where do we start? I’ve got you covered. This week, we’re kicking things off with the two baby Algebra upgrades that make the absolute biggest difference. These make everything else I’ll teach afterward possible. So listen up!

Baby Algebra Upgrade #1: Recognize what constitutes a linear equation in the first place.

As you may recall, a line (or linear equation) is simply an equation where both y and x are raised to the first power. In other words, there is no y2 or x3 or sin(x) or 1/x. Just y and x, maybe multiplied by a number, and everything is a different term (i.e. added or subtracted together).

Here’s the most popular form of a line:

y = mx + b

This way of representing the line is called “slope-intercept form,” where “m” is the slope and “b” is the y-intercept.

But these are also legit:

y – y1 = m(x – x1)
This is “point-slope form,” where (x1, y1) is a point on the line and “m” is the slope.

Ax + By = C

This is “standard form,” where “A,” “B,” & “C” are constants, usually integers.

Notice: in all of these examples, x and y are never multiplied or divided together. They might be multiplied or divided by a number and then added or subtracted after that.

For example, these equations are NOT lines:

x/y = 14

2xy + x = 5

But these ARE:

y = 3x – 5

y + 1 = 2(x – 4)

4x – 3y = 9

So now that you get what makes a line in the first place, here’s your trick:

If you need to find out the slope or the y-intercept, just take the equation they gave you and solve for y, putting the equation in the format y = mx + b!

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Need to know which line in the answer choices is parallel to (i.e. has the same slope as) the line equation given? Just solve it for y and you’ll see your slope next to the x!

Need to know where the equation will cross the y-axis on the coordinate plane? Just solve it for y and you’ll see your y-intercept as the random number that's being added or subtracted and is NOT multiplied by x.

Knowing what you’re looking at and how to work with it is the foundation of all of the upgrades we’re doing in this series, so pat yourself on the back if you’ve mastered this one, and take a moment to go back and reread if you’re not quite sure yet. If you are…let’s go on to the next one!

Baby Algebra Upgrade #2: Understand the real-world interpretations of a line.

This is my favorite, because it often blows students’ minds.

Say you have a scenario as follows: the wind speed (s) in miles per hour on a mountain relates to the height above sea level (h) in feet with the equation:

s = 3h + 10

What does the 10 signify? What about the 3?

If you’re like most of my students, you’d say, “Kristina, the 10 is the y-intercept and the 3 is the slope!”

To which I’d say, “Okay, but what does that mean?” and get…crickets.

Y-Intercept:

“Even if my ____ (x) were 0, my ____ (y) would still be ____ (y-intercept).”

In this case:

“Even if my height above sea level were 0, my wind speed would still be 10.”

Slope:

“For every ____ (unit of x), my ____ (y) increases/decreases by ____ (m).”

In this example:

“For every foot above sea level, my wind speed increases by 3.”

Make sense?

Here’s the best part: the SAT always asks these questions, in pretty much exactly this way. They’ll literally test whether you know what the numbers in a linear equation refer to—so now you have it in your back pocket!

You’re welcome. ;)

So that’s the foundation you need to upgrade your skills with linear equations!

This is literally the tip of the iceberg of upgrading simple Algebra skills, but I felt they were so relevant that even sharing a little bit would help out. If you need the depth and individuality that I can only offer if we’re working one on one, though, you can contact me here.

In my next post, I’ll be showing you what you can do with this upgraded foundation…so stay tuned!