Perfect1600

Digital SAT · Solve it in Desmos

How to Solve Quadratic-Linear Systems on Desmos (Digital SAT)

For a count of solutions, graphing turns the question into simple counting, and for coordinates it beats the algebra outright. Solve by hand only when the numbers are tidy or you want to confirm an exact value.

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

Full method for quadratic-linear systems

The Desmos steps

  1. 1

    Graph both

    Type the line and the curve as y = ... on their own lines in Desmos.

  2. 2

    Click each crossing

    Desmos labels every intersection, and the number of crossings is the number of solutions.

  3. 3

    Read what is asked

    Take the coordinate or the count straight off the graph.

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

Worked in Desmos

Easy example
The graphs of x=3x = 3 and y=2x2−5y = 2x^{2} - 5 intersect at the point (x,y)(x, y) in the xy-plane. What is the value of yy?
A
11
1313
C
1818
D
3131

A: Incorrect. This computes 2(3)−52(3) - 5, treating x2x^{2} as xx.

B: Correct. Substituting x=3x = 3 gives y=2(3)2−5=18−5=13y = 2(3)^{2} - 5 = 18 - 5 = 13.

C: Incorrect. This computes 2(3)22(3)^{2} but forgets to subtract 5.

D: Incorrect. This computes (2⋅3)2−5(2 \cdot 3)^{2} - 5, squaring after multiplying by 2.

Explanation

Substitute x=3x = 3: y=2(3)2−5=18−5=13y = 2(3)^{2} - 5 = 18 - 5 = 13.

Medium example
How many points of intersection do the graphs of y=2x+1y = 2x + 1 and y=x2+5y = x^{2} + 5 have?
A
Exactly one
B
Exactly two
C
Infinitely many
Zero

A: Incorrect. Exactly one would require the discriminant to equal 0, but here it is negative.

B: Incorrect. Exactly two would require a positive discriminant, but here it is negative.

C: Incorrect. The graphs are a line and a parabola, which cannot overlap entirely.

D: Correct. Setting x2+5=2x+1x^{2} + 5 = 2x + 1 gives x2−2x+4=0x^{2} - 2x + 4 = 0, with discriminant 4−16<04 - 16 < 0, so there are no real intersections.

Explanation

The equation x2−2x+4=0x^{2} - 2x + 4 = 0 has discriminant 4−16=−12<04 - 16 = -12 < 0, so there are zero intersections.

Hard example
The system y=x2−2xy = x^{2} - 2x and y=x+4y = x + 4 is graphed in the xy-plane. Which ordered pair is a solution?
(4,8)(4, 8)
B
(2,6)(2, 6)
C
(3,7)(3, 7)
D
(−1,5)(-1, 5)

A: Correct. At (4,8)(4, 8): 8=4+48 = 4 + 4 and 8=42−2(4)8 = 4^{2} - 2(4), satisfying both equations.

B: Incorrect. (2,6)(2, 6) satisfies y=x+4y = x + 4 but not the quadratic, since 4−4=04 - 4 = 0.

C: Incorrect. (3,7)(3, 7) satisfies y=x+4y = x + 4 but not the quadratic, since 9−6=39 - 6 = 3.

D: Incorrect. (−1,5)(-1, 5) satisfies neither, since −1+4=3-1 + 4 = 3 and 1+2=31 + 2 = 3.

Explanation

Setting x2−2x=x+4x^{2} - 2x = x + 4 gives x2−3x−4=0x^{2} - 3x - 4 = 0, so x=4x = 4 or x=−1x = -1; the point (4,8)(4, 8) is listed.

Detailed explanation

Setting the expressions equal gives x2−2x=x+4x^{2} - 2x = x + 4, or x2−3x−4=0x^{2} - 3x - 4 = 0. Factoring (x−4)(x+1)=0(x - 4)(x + 1) = 0 gives x=4x = 4 or x=−1x = -1, with points (4,8)(4, 8) and (−1,3)(-1, 3). Only (4,8)(4, 8) appears; (−1,5)(-1, 5) uses the wrong y-value.

Try it with the calculator

Question 1easy

The system y=x+2y = x + 2 and y=x2y = x^{2} is graphed in the xy-plane. Which ordered pair (x,y)(x, y) is a solution to this system?

Question 2easy

The graphs of x=3x = 3 and y=x2+1y = x^{2} + 1 intersect at the point (x,y)(x, y) in the xy-plane. What is the value of yy?

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 quadratic-linear systems questions on the SAT?

For a count of solutions, graphing turns the question into simple counting, and for coordinates it beats the algebra outright. Solve by hand only when the numbers are tidy or you want to confirm an exact value.

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 quadratic-linear systems, 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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