Perfect1600

Digital SAT · Solve it in Desmos

How to Solve Number of Solutions of an Equation on Desmos (Digital SAT)

Graphing both sides shows the count directly, so you never have to interpret matching terms by hand. Simplify algebraically when 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 an equation

The Desmos steps

  1. 1

    Graph each side

    Type y equals the left side and y equals the right side into Desmos.

  2. 2

    Read the relationship

    One crossing is one solution; parallel lines are none; the same line is infinitely many.

  3. 3

    Test a constant

    For an unknown constant, try values and watch when the two lines coincide or split apart.

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 does the equation 6x−3=6x−86x - 3 = 6x - 8 have?
A
Exactly one
Zero
C
Exactly two
D
Infinitely many

A: Incorrect. A single solution would require the variable to survive after simplifying.

B: Correct. Subtracting 6x6x leaves −3=−8-3 = -8, which is false, so there are no solutions.

C: Incorrect. A linear equation cannot have exactly two solutions.

D: Incorrect. Infinitely many solutions would require a true statement after the variable cancels.

Explanation

Subtract 6x6x from both sides: −3=−8-3 = -8. This is never true, so there is no solution.

Medium example
How many solutions does the equation 2(3x+5)=6x−12(3x + 5) = 6x - 1 have?
Zero
B
Exactly one
C
Exactly two
D
Infinitely many

A: Correct. Distributing gives 6x+10=6x−16x + 10 = 6x - 1; subtracting 6x6x leaves 10=−110 = -1, which is false.

B: Incorrect. The variable cancels, so it cannot have a single solution.

C: Incorrect. A linear equation cannot have exactly two solutions.

D: Incorrect. Infinitely many solutions would require a true statement after the variable cancels.

Explanation

Distribute: 6x+10=6x−16x + 10 = 6x - 1. Subtract 6x6x: 10=−110 = -1, which is never true, so no solution.

Hard example
2(kx−3)=8x+52(kx - 3) = 8x + 5 In the given equation, kk is a constant. The equation has no solution. What is the value of kk?
A
22
44
C
88
D
1616

A: Incorrect. This divides the coefficient 8 by 4 instead of by 2.

B: Correct. 2kx−6=8x+52kx - 6 = 8x + 5; no solution needs 2k=82k = 8 (and −6≠5-6 \neq 5), so k=4k = 4.

C: Incorrect. This uses the coefficient 8 directly instead of solving 2k=82k = 8.

D: Incorrect. This multiplies 2×82 \times 8 instead of dividing.

Explanation

Expand: 2kx−6=8x+52kx - 6 = 8x + 5. No solution requires equal variable coefficients with unequal constants: 2k=82k = 8, so k=4k = 4.

Detailed explanation

Distributing gives 2kx−6=8x+52kx - 6 = 8x + 5. The equation has no solution when the variable coefficients match but the constants differ. Since −6≠5-6 \neq 5, set 2k=82k = 8, giving k=4k = 4.

Try it with the calculator

Question 1easy

How many solutions does the equation 2x+5=2x+92x + 5 = 2x + 9 have?

Question 2easy

How many solutions does the equation 3x+7=3x+73x + 7 = 3x + 7 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 an equation questions on the SAT?

Graphing both sides shows the count directly, so you never have to interpret matching terms by hand. Simplify algebraically when 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 an equation, 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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