Digital SAT · Linear Equations & Inequalities

How to Solve Number of Solutions of an Equation on the Digital SAT

A single equation can have one solution, none, or infinitely many, and the tell is in the two sides once you simplify. Different variable terms give exactly one solution. Matching variable terms with different constants give none, since the sides can never be equal. Identical sides give infinitely many, since every value works. Graphing each side as its own line in Desmos shows the same thing.

Written from Perfect1600’s analysis of every number of solutions of an equation question in our bank·Method checked against the current Bluebook test

per test: less than 1 per test
typical difficulty: Medium
practice questions: 27

How we counted

Frequency reflects how often this question type appears on a full-length Digital SAT. The difficulty mix reflects every question of this type across our bank.

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What the question bank shows

1, 0, or ∞

Compare the two sides

Different variable terms give one; matching terms decide none or infinitely many.

85%

Graph each side

Most are read by graphing both sides and seeing how the lines relate.

the shift

Structure, not x

The answer is about the two sides matching, not about solving for x.

How to recognize number of solutions of an equation questions

A single equation is given, often with the variable on both sides.
The question asks how many solutions, or for a constant that forces none or infinitely many.
The words no solution, infinitely many, or exactly one appear.
Choices are counts or values of a constant.

Why students miss these

The instinct to solve for x misses the point, since the answer is structural. After simplifying, matching variable terms with different constants gets read as one solution instead of none. The constant version adds algebra: you make the variable coefficients match for none or infinitely many, then check the constants. Simplify fully and compare the two sides.

The step-by-step method

1
Simplify both sides
Distribute and combine like terms so each side is as simple as possible.
2
Compare the variable terms
Different variable terms mean exactly one solution.
3
Compare the constants if variables match
Equal variable terms with different constants give none; identical sides give infinitely many.
4
Solve for a constant
Set the variable coefficients equal for none or infinitely many, then check the constants.

Solving it on Desmos

Graph each side. Type y equals the left side and y equals the right side into Desmos.
Read the relationship. One crossing is one solution; parallel lines are none; the same line is infinitely many.
Test a constant. For an unknown constant, try values and watch when the two lines coincide or split apart.

Full Desmos walkthrough for number of solutions of an equation→

When to use it: 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.

See it live: open a real question in the same Desmos calculator you get on Perfect1600, with the equations already typed in.

Worked examples

Easy example

How many solutions does the equation

6x - 3 = 6x - 8

have?

Exactly one

Zero

Exactly two

Infinitely many

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

B: Correct. Subtracting $6x$ leaves $-3 = -8$ , 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 $6x$ from both sides: $-3 = -8$ . This is never true, so there is no solution.

Medium example

How many solutions does the equation

2(3x + 5) = 6x - 1

have?

Zero

Exactly one

Exactly two

Infinitely many

A: Correct. Distributing gives $6x + 10 = 6x - 1$ ; subtracting $6x$ leaves $10 = -1$ , 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: $6x + 10 = 6x - 1$ . Subtract $6x$ : $10 = -1$ , which is never true, so no solution.

Hard example

2(kx - 3) = 8x + 5

In the given equation,

k

is a constant. The equation has no solution. What is the value of

k

2

4

8

16

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

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

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

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

Explanation

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

Detailed explanation

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

The common traps

Pattern	What it does	The tell
Solved instead of compared	Tried to find x when the answer is a count.	Simplify both sides and compare structure rather than solving.
None versus infinitely many	Read matching variable terms with different constants as one solution.	Same variable term, different constant is none; identical sides is infinitely many.
Constant condition wrong	Set the wrong coefficients equal for the special case.	Match the variable coefficients first, then check the constants.

Try it: two real questions

Question 1easy

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

Question 2easy

How many solutions does the equation $3x + 7 = 3x + 7$ 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.

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Often confused with

Linear Equations & Inequalities

Linear Equations

Systems of Linear Equations

Number of Solutions of a System

Equivalent Expressions

Literal Equations

Common questions

How do you tell if a linear equation has no solution?

Simplify both sides. If the variable terms match but the constants differ, the equation is never true, so no solution. Graphing shows two parallel lines.

When does an equation have infinitely many solutions?

When both sides simplify to the same expression, so every value works. On a graph, the two sides are the same line.

How do I find the constant that gives infinitely many solutions?

Make the variable coefficients equal on both sides, then the constants equal too. If only the coefficients match but the constants differ, you get no solution instead.

Can Desmos count solutions of an equation?

Yes. Graph each side as a line; one crossing is one solution, parallel lines are none, the same line is infinitely many. It avoids misreading the simplified terms.

Practice number of solutions of an equation the way it is tested

Start with the free 16-question diagnostic, then drill this type with step-by-step reasoning on every question.

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What the question bank shows

How to recognize number of solutions of an equation questions

Why students miss these

The step-by-step method

Simplify both sides

Compare the variable terms

Compare the constants if variables match

Solve for a constant

Solving it on Desmos

Worked examples

The common traps

Try it: two real questions

Question 3 is ready when you are

Often confused with

Related reading

Common questions

How do you tell if a linear equation has no solution?

When does an equation have infinitely many solutions?

How do I find the constant that gives infinitely many solutions?

Can Desmos count solutions of an equation?

Practice number of solutions of an equation the way it is tested