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Digital SAT · Solve it in Desmos

How to Solve Intercepts on Desmos (Digital SAT)

Graphing is quickest for fraction or standard forms where the algebra is fiddly, and it removes the axis-swap error. Read by hand only when the equation is already in slope-intercept form.

Perfect1600 Content Team·Uses the same Desmos calculator that is built into the Digital SAT

Full method for intercepts

The Desmos steps

  1. 1

    Graph the equation

    Type the line in any form to see it plotted.

  2. 2

    Click the crossings

    The x-axis crossing for the x-intercept, the y-axis crossing for the y-intercept; each is labeled.

  3. 3

    Take the value asked

    Read the coordinate the question wants from the labeled crossing.

Open a real question in the same Desmos calculator you get on Perfect1600, with the setup already typed in:

Worked in Desmos

Medium example
A line passes through the points (0,−3)(0, -3) and (2,5)(2, 5) in the xy-plane. What is the y-coordinate of the y-intercept of the line?
−3-3
B
11
C
44
D
55

A: Correct. The point (0,−3)(0, -3) is on the y-axis, so the y-intercept is −3-3.

B: Incorrect. This averages the two y-values instead of reading x=0x = 0.

C: Incorrect. This is the slope of the line, not the y-intercept.

D: Incorrect. This is the y-value at x=2x = 2, not at x=0x = 0.

Explanation

The y-intercept is where x=0x = 0; the point (0,−3)(0, -3) gives y=−3y = -3.

Medium example
The equation 5x+2y=205x + 2y = 20 represents a line in the xy-plane. What is the x-coordinate of the x-intercept of the line?
A
−4-4
44
C
1010
D
2020

A: Incorrect. This has the wrong sign for the x-intercept.

B: Correct. Setting y=0y = 0 gives 5x=205x = 20, so x=4x = 4.

C: Incorrect. This is the y-coordinate of the y-intercept, not the x-intercept.

D: Incorrect. This uses the constant 20 without dividing by 5.

Explanation

The x-intercept occurs at y=0y = 0: 5x=205x = 20, so x=4x = 4.

Hard example
A line in the xy-plane is defined by x5−y2=1\frac{x}{5} - \frac{y}{2} = 1. What is the x-coordinate of the x-intercept of the line?
A
−2-2
B
11
C
22
55

A: Incorrect. This is the y-coordinate of the y-intercept, not the x-intercept.

B: Incorrect. This is the right side of the equation, not the intercept.

C: Incorrect. This uses the denominator under yy instead of under xx.

D: Correct. Setting y=0y = 0 gives x5=1\frac{x}{5} = 1, so x=5x = 5.

Explanation

The x-intercept occurs at y=0y = 0: x5=1\frac{x}{5} = 1, so x=5x = 5.

Detailed explanation

Setting y=0y = 0 leaves x5=1\frac{x}{5} = 1, so x=5x = 5. The denominator under xx gives the x-intercept directly.

Try it with the calculator

Question 1hard

A line in the xy-plane has a slope of 3 and passes through the point (2,5)(2, 5). What is the y-coordinate of the point where the line crosses the y-axis?

Question 2hard

A line in the xy-plane has a slope of −2-2 and passes through the point (3,4)(3, 4). What is the x-coordinate of the x-intercept of the line?

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

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Desmos questions

Can Desmos solve intercepts questions on the SAT?

Graphing is quickest for fraction or standard forms where the algebra is fiddly, and it removes the axis-swap error. Read by hand only when the equation is already in slope-intercept form.

Is the Desmos calculator really built into the Digital SAT?

Yes. The Bluebook testing app includes the Desmos graphing calculator on every Math question, so the method here is one you can use on test day, not a workaround.

Should I still learn the algebra?

Yes. Desmos is fastest when you know what to type and what to read. Learn the underlying method on the full guide, then use the calculator to move quickly and to check your work.

Master intercepts, calculator and all

Every Digital SAT question type comes with the method, the Desmos shortcut, and free practice with instant feedback.

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