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SAT Slope From Two Points

Given two points, the SAT asks for the slope, for the equation of the line through them, or for a third point that lies on the same line. The slope formula is short, and the mistakes are almost all about order: keeping the two points in the same order in the numerator and the denominator.

The rule

Slope is the difference in y-values divided by the difference in x-values, with both differences taken in the same order.

The slope is the change in y divided by the change in x. For the points with coordinates 1 and 4 and coordinates 5 and 16, subtract in a consistent order: 16 minus 4 is 12 on top, and 5 minus 1 is 4 on the bottom, so the slope is 3. If you had subtracted the other way you would get negative 12 over negative 4, which is still 3. What breaks the answer is mixing the orders, taking 16 minus 4 over 1 minus 5.

Once you have the slope, build the equation by substituting one point. With slope 3 and the point with coordinates 1 and 4, write 4 equals 3 times 1 plus b, so b equals 1 and the line is y equals 3x plus 1. Verify against the other point: 3 times 5 plus 1 is 16, which matches. That verification step is free and catches sign errors.

Negative slopes need attention with the signs. For the points with coordinates negative 2 and 7 and coordinates 4 and negative 5, the numerator is negative 5 minus 7, which is negative 12, and the denominator is 4 minus negative 2, which is 6. The slope is negative 2. Subtracting a negative is the most common place to lose a sign, so write the double negative out rather than doing it in your head.

Reading points off a graph works the same way, but pick lattice points where the line clearly crosses grid intersections. Estimating from a point that sits between gridlines introduces error the answer choices are usually too close together to absorb.

Collinearity questions are the same skill asked sideways. If three points lie on one line, then the slope between the first two must equal the slope between the second and third. So to find an unknown coordinate, set those two slope expressions equal to each other and solve. That single equation replaces any need to find the full line.

The trap

Subtracting the y-values in one order and the x-values in the other, which flips the sign of the slope.

Try it

Try one
Real question

Which is NOT on the line y = 2x - 3?

A laptop battery's use satisfies 9v + 4b = 468, where v is the number of minutes of video played and b is the number of minutes of background work. If 32 minutes of video are played on one charge, how many minutes of use in all does that charge give?

  • A) 32
  • B) 45
  • C) 77 ✓
  • D) 117

Why: The video uses 9(32) = 288, leaving 180 for background work, which is 180/4 = 45 minutes. The charge gives 32 + 45 = 77 minutes of use. Choice B counts only the background minutes. Answering 32 means you gave the video minutes only. Answering 45 means you gave the background minutes only. Answering 117 means you divided 468 by 4 without subtracting the video's share first.

Practice more of these free

Questions students ask

Does it matter which point I call the first one?

No, as long as you are consistent. Using either point first gives the same slope, because both the numerator and denominator change sign together. The error appears only when you start with one point on top and the other on the bottom.

What does a slope of zero mean compared to undefined slope?

A slope of zero means the two points have the same y-value, so the line is horizontal. An undefined slope means the two points have the same x-value, which makes the denominator zero, so the line is vertical and cannot be written in slope-intercept form.

Guides that cover this

More on linear equations in two variables