Desmos for the SAT · Technique 4 of 23

Read a Line's Slope and Intercepts from Its Graph

Slope, x-intercept, y-intercept: three answers, one graph.

SolvesSlopeInterceptsEquation of a line

Watch the lesson

The whole move worked on one real SAT question, narrated.

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A second example, another equation with x and y on both sides, worked end to end: both crossings clicked, then divided for the slope.

Linear function questions on the SAT almost always come down to one of three things: the slope, the y-intercept, or the x-intercept. Once you know which one they want, the rest is quick.

If the equation is already solved for y, you can often read the slope and y-intercept straight off the numbers. When they want the x-intercept, or the equation shows up in a messier form, graphing it and clicking the axis crossing is the fastest way to a clean answer.

Word problems rename the same two numbers, so watch for that. The slope m is the rate of change, how fast something grows per step, and the y-intercept b is the initial value or starting cost, the amount before anything changes. Decide which of the two the question wants, then read it from $y = mx + b$ as usual.

Nova

Slope, y-intercept, x-intercept: linear questions ask you the same three things over and over, so this one move answers most of them. Graph the line, click the crossing, read the number, done. No solving for x under time pressure.

Now you try it

The rest of the page takes that move slowly. First how it compares with doing the algebra, then the steps in order, then a real SAT question with a live Desmos calculator to try it in.

The algebra way vs the Desmos way

This technique sidesteps rearranging a fractional equation into $y = mx + b$ .

The question

(3x)/7 = -(5y)/9 + 21

What is the slope of the line?

The algebra way

Clear both fractions first: multiply through by 63, giving 27x = -35y + 1323.
Solve for y: y = -(27/35)x + 37.8. Only now is the slope visible, and the sign, the reciprocal and the division are each a place to slip.

The Desmos way

Type (3x)/7 = -(5y)/9 + 21 as given and select it: dots appear on both axis crossings.
Click each dot for (0, 37.8) and (49, 0), then type (37.8-0)/(0-49) and press the fraction toggle: -27/35.

The payoff: Two clicks and one division, with nothing rearranged and no fraction chain to trust. The two intercepts come along for free if a question asks for those instead.

Step by step

The whole move, all 5 steps

Type the equation exactly as given, even with x and y on opposite sides.
Click inside the equation box to reveal the gray dots. Desmos marks both axis crossings on its own, so there is no need to add x = 0 or y = 0 as extra lines.
Click the dot on the y-axis for the y-intercept, and the dot on the x-axis for the x-intercept.
For the slope, use two points on the line: type (second y - first y)/(second x - first x), keeping the points in the same order on top and underneath, and let Desmos divide. The two intercepts are the natural pair. If the line passes through the origin they are the same point, so type x = 1 on its own line and click where the two lines cross for a second point. Press the fraction button if the answer choices are fractions.
Answer with the one of the three the question asked for. If it asks for the sum or product of the intercepts, read both dots and combine them.

The question

The equation $\frac{3x}{7}=-\frac{5y}{9}+21$ defines a line in the xy-plane.

What is the slope of the line?

A-35/27
B-27/35
C27/35
D3/7

Step 1 of 5

Type the equation exactly as given: (3x)/7 = -(5y)/9 + 21. Press the right arrow after each denominator, the 7 and the 9, to step back out of the fraction before typing on. Desmos graphs it with x and y on opposite sides, so there is nothing to rearrange first.

In the calculator: (3x)/7→=-(5y)/9→+21

This is the bit worth noticing: the equation is not solved for y, and you still do not have to solve it.

Syntax: Type the equation exactly as printed, even with x and y on both sides; Desmos still draws the line. The gray dots appear only once the expression is selected, so click inside the equation box first. For the slope, type the division as (y-intercept - 0)/(0 - x-intercept), then press the fraction button at the left of that line if the choices are fractions.

Now try it on a real SAT question

Head or scratch paper. Already in $y = mx + b$ , so the y-intercept is the number on the end. Read it and move on. This is the case the lesson warns about: graphing it is slower than reading it.

The function f is defined by $f(x) = \frac{1}{5}x + 3$ .

What is the y-intercept of the graph of y = f(x) in the xy-plane?

Pick an answer to get instant coaching from Nova.

Stuck?

Your turnstarts empty

Set this question up yourself, then read the answer straight off the graph.

Worked it out on scratch paper? Type it into Desmos anyway and check. It takes a couple of seconds and it catches the slip you cannot spot by re-reading your own work.

The Desmos trap: when to close the calculator

If the equation is already solved for y and the question only asks for slope or y-intercept, read the coefficients directly from $y = mx + b$ . No graph needed. This lesson uses a question where that shortcut is unavailable, which is exactly when you want the graph.
For a word problem that only asks for a rate of change between two given points, use the slope formula on those two points instead of graphing the whole line.

All Desmos techniques

Put this move into real practice

"Read a Line's Slope and Intercepts from Its Graph" shows up all over the real SAT. SAT Prep Quest drills the skill at your level, so reaching for it becomes automatic by test day.

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