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Digital SAT · Solve it in Desmos

How to Solve Number of Solutions of a System on Desmos (Digital SAT)

Graphing makes the relationship visible at once, so counting solutions is just looking. Compare slopes by hand when the equations are already in slope-intercept form or a constant must be solved exactly.

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

Full method for number of solutions of a system

The Desmos steps

  1. 1

    Graph both equations

    Type both lines as y equals expressions.

  2. 2

    See how they relate

    Crossing once is one solution; parallel and apart is none; overlapping is infinitely many.

  3. 3

    Test a constant

    For an unknown constant, try values and watch when the lines become parallel or identical.

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

Worked in Desmos

Easy example
How many solutions (x,y)(x, y) does the system of equations y=4x+1y = 4x + 1 and y=4x+7y = 4x + 7 have?
Zero
B
Exactly one
C
Exactly two
D
Infinitely many

A: Correct. Both slopes are 4 but the intercepts differ (1 and 7), so the lines are parallel and never meet.

B: Incorrect. Exactly one solution requires different slopes.

C: Incorrect. Two lines cannot intersect at exactly two points.

D: Incorrect. Infinitely many solutions require identical intercepts too.

Explanation

Both lines have slope 4 but different y-intercepts (1 and 7), so they are parallel and have zero solutions.

Medium example
2x−5y=82x-5y=8 and −6x+15y=−24-6x+15y=-24. How many solutions does this system of equations have?
A
Zero
B
Exactly one
C
Exactly two
Infinitely many

A: Incorrect. Zero solutions would require parallel lines with different constants, but the second equation is −3-3 times the first.

B: Incorrect. Exactly one solution requires different slopes, but the equations are scalar multiples.

C: Incorrect. Two distinct lines can intersect at most once; here the lines are identical.

D: Correct. Multiplying the first equation by −3-3 gives −6x+15y=−24-6x+15y=-24, exactly the second equation, so they are the same line: infinitely many solutions.

Explanation

The second equation is -3 times the first, so they describe the same line: infinitely many solutions.

Hard example
For what value of kk does the system kx+2y=5kx+2y=5 and 3x+y=43x+y=4 have no solution?
A
33
66
C
99
D
1212

A: Incorrect. k=3k=3 does not make the slopes equal.

B: Correct. No solution requires equal slopes: −k2=−3-\frac{k}{2}=-3, so k=6k=6 (and the lines stay distinct).

C: Incorrect. k=9k=9 gives slope −4.5-4.5, not −3-3.

D: Incorrect. k=12k=12 gives slope −6-6, not −3-3.

Explanation

No solution needs equal slopes: -k/2 = -3, so k = 6.

Try it with the calculator

Question 1easy

How many solutions (x,y)(x, y) does the system of equations y=2x+3y = 2x + 3 and y=5x−1y = 5x - 1 have?

Question 2easy

How many solutions (x,y)(x, y) does the system of equations y=−x+2y = -x + 2 and 2y=−2x+42y = -2x + 4 have?

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.

Start the free diagnostic

Desmos questions

Can Desmos solve number of solutions of a system questions on the SAT?

Graphing makes the relationship visible at once, so counting solutions is just looking. Compare slopes by hand when the equations are already in slope-intercept form or a constant must be solved exactly.

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 number of solutions of a system, 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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