While the SAT math section includes a handful of useful formulas, all of these “freebies” are directly related to geometry. There remain a significant number of important equations that you’ll need to know on your own to successfully navigate the math section of the test. In this blog post, we’ll be looking at what equations you’ll need to know about graphing lines.
Equation of a Line: y = mx + b
While you’ve no doubt heard this equation for years, if it’s been awhile since you’ve last used it, it’s worth a quick review before the big test day. Remember:
m = slope of the line
b = the y-intercept for the line (the place on the y-axis where the line you’re graphing hits it)
Note that when the line runs through the origin (0, 0), then the origin is the y-intercept and your equation will look more like y = mx.
If you encounter an equation for a line that isn’t in this format, rearrange it until it is. The SAT loves to check your familiarity with this equation by asking seemingly simple questions (whether the slope is positive or negative, what’s the y intercept, etc.) but coupled with an equation that’s been rearranged just enough to make things tricky, such as b = mx – y. By quickly putting it into y = mx + b format, you’ll be able to identify useful information about the line without actually graphing it.
Slope formula
Of course, sometimes instead of being given an equation for a line, you’ll be provided two points and expected to come up with the equation yourself. To do this you’ll first need to find the slope.
Remember, slope is equal to the vertical change over the horizontal change. If you’ve been given two points on the line,
A(x1,y1)
B(x2,y2)
then the formula for finding the slope is:
(y2 – y1)
(x2 – x1)
Once you have the slope, solve your line equation y = mx + b for b using one of the provided coordinates.
Midpoint and Length
Sometimes an SAT question will want to know the midpoint of a line connecting two points. If our two points are once again:
A(x1,y1)
B(x2,y2)
then the midpoint is found using this formula:
((x2 + x1), (y2 + y1))
2 2
Rather than memorizing this equation, though, it may be easier to think of it as taking the average of the two values for x and the average for the two values of y to generate values for the midpoint.
Similarly, sometimes you may also be asked the distance between two points. Rather than memorizing the equation
distance = sqrt((x2−x1)^2+(y2−y1)^2)
instead, focus on thinking about the two coordinates as vertices of a right triangle connected by the hypotenuse. (You may even want to actually graph it that way to help you visualize it during the test.) You can then find the distance between them using the pythagorean theorem, which happens to be one of the included equations on the exam.
You can expect to encounter at least a couple of questions related to lines on every SAT exam. Since the formulas for working with lines aren’t provided, it’s important to take some time beforehand to review the basics to make sure you have things all lined up before the big day.