Question type

Number of solutions questions on the SAT

A linear equation has no solution when the x terms match but the constants don't, infinitely many when both sides are identical, and exactly one otherwise. For a system, scale one equation until the x terms match: same y terms with different right sides means no solution; everything equal means infinitely many.

These questions give you a linear equation, or a system of two, with a constant in it, and ask when it has no solution, exactly one solution, or infinitely many. A few ask how many solutions there are. Most ask for the value of the constant that makes it happen.

It appeared on all five official digital SAT practice tests we analysed, seven times in all, so expect at least one on test day.

On 5 of 5 official practice tests
59 practice questions of this type
Desmos helps

Why students miss it

The reflex is to solve for $x$ . That goes nowhere, because the question is about the shape of the equation, not its answer: once the $x$ terms cancel, there is nothing left to solve.

In a system, the trap is the scale factor. If the second equation is the first one doubled, the scale factor is 2, but the question asks for the constant that doubling produces. Choosing 2 is the most common wrong answer.

And same slope is not the whole story. Two lines with the same slope are parallel (no solution) if their constants differ, and the same line (infinitely many) if they don't.

How to spot it

The words "no solution", "infinitely many solutions" or "exactly one solution".
A letter described as "a constant", such as $k$ , $a$ or $c$ , beside $x$ or alone on one side.
A system where one equation looks like a multiple of the other.

The method

One equation

Collect the terms so it reads $a x + b = c x + d$ , then compare the two sides.

The $x$ coefficients differ ( $a \neq = c$ ): exactly one solution.
The $x$ coefficients match, the constants don't ( $a = c$ , $b \neq = d$ ): no solution.
Both sides are identical ( $a = c$ , $b = d$ ): infinitely many solutions.

A system of two equations

Multiply one equation so its $x$ term matches the other's, then compare the rest.

The $y$ terms match, the right sides don't: no solution: parallel lines.
Everything matches: infinitely many: the same line twice.
The $y$ terms differ: exactly one solution: the lines cross once.

With Desmos

Desmos turns the constant into a slider, so you can watch the lines move until they never meet.

Type the first equation exactly as written, for example kx+4y=10. Desmos sees the k and offers a slider: tap "add slider".
Type the second equation. For a single equation, type each side as its own line: y=8x+c and y=4(2x-5)+7.
Drag the slider until the lines are parallel (no solution) or lie on top of each other (infinitely many).
Tap the number on the slider and type the value you found, to check it exactly rather than by eye.

Desmos is quickest for "no solution", where parallel lines are easy to see. For "infinitely many", two lines that are almost on top of each other look identical, so confirm the value with the algebra above.

Two real questions

From the 1500prep bank, the way the app shows them. Answer first, then open the explanation.

1Question 1 of 2#611Alg

kx + 4y = 10 3x + 2y = 7 In the system of equations above, k is a constant. If the system has no solution, what is the value of k?

Pick an answer, then check it.

Show the answer and explanation

No solution means the two lines are parallel: the

x

- and

y

-coefficients scale by the same factor, but the constants don't. Going from the second equation to the first,

4 y

2

times

2 y

, so multiply everything by

2

3 x

becomes

6 x

. So

k = 6

. (Check:

7 \cdot 2 = 14 \neq = 10

, so the lines really are parallel, not identical.)

2Question 2 of 2#959Alg

In the equation 8x+c=4(2x−5)+7, c is a constant. If the equation has infinitely many solutions, what is the value of c?

Pick an answer, then check it.

Show the answer and explanation

Infinitely many solutions means both sides are the same expression. Expand the right:

4 (2 x - 5) + 7 = 8 x - 20 + 7 = 8 x - 13

. The

8 x

terms already match, so

c = - 13

Questions students ask

How do you know when a system of equations has no solution on the SAT?

When the two lines have the same slope but different y-intercepts. In standard form, scale one equation so the x terms match: if the y terms match too but the right-hand sides differ, the system has no solution.

When does an equation have infinitely many solutions?

When both sides simplify to exactly the same expression, so every value of x works. For a system, it is when one equation is a multiple of the other, right-hand side included.

Can I use Desmos for number of solutions questions?

Yes. Type the equation with its constant, add a slider, and move it until the lines are parallel or overlap. Check the exact value afterwards, because nearly overlapping lines look identical.

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