If I get x = 0 when I solve, does that mean no solution?

No. If your answer is x = 0, that is one solution. Zero is a real number, and it is the value that makes the equation true. 'No solution' means that no value at all works—you get that when the equation simplifies to a false statement like 5 = 3 or 0 = 2.

How can I tell the difference between 'no solution' and 'infinitely many solutions'?

After you simplify and the variable cancels, look at what remains. If you get a true statement (like 0 = 0 or 5 = 5), you have infinitely many solutions. If you get a false statement (like 0 = 3 or 5 = 2), you have no solution. The variable cancels in both cases, but the number statement that's left tells you which one it is.

Why does the variable sometimes disappear when I solve?

The variable disappears when the expression on both sides of the equation is actually the same underneath, or fundamentally incompatible. If they're the same (an identity like 2x + 6 = 2(x + 3)), then the equation is always true. If they're incompatible (a contradiction like 2x + 6 = 2x + 5), then the equation is never true. When you solve, you're revealing which one it is.

Can I check my answer by substituting a number if the equation has infinitely many solutions?

Yes. If the equation has infinitely many solutions, you can pick any real number and substitute it in, and it will make both sides equal. For example, in 2x + 6 = 2(x + 3), try x = 10: left side is 2(10) + 6 = 26, right side is 2(10 + 3) = 26. It works. Try x = -5: left side is 2(-5) + 6 = -4, right side is 2(-5 + 3) = -4. It works again. This confirms infinitely many solutions.

M8MATH-4.4

One, None or Infinitely Many Solutions

Learn to solve linear equations and determine whether they have exactly one solution, infinitely many solutions, or no solution.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on One, None or Infinitely Many Solutions, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

When you solve an equation, you're looking for the value of

x

that makes it true. But not every equation has exactly one answer. Some equations are always true no matter what

x

is, some are never true, and some have exactly one

x

that works. In this lesson, you'll learn how to transform equations into forms that reveal how many solutions they have, and why that happens. This skill is essential for understanding linear equations deeply and for spotting patterns in algebra.

The Three Cases: One Solution, No Solution, Infinitely Many

When you solve a linear equation by transforming it (using inverse operations on both sides), one of three things happens.

One Solution: The equation simplifies to a statement like

x = 5

. This means exactly one value of

x

makes the equation true. For example,

2x + 3 = 11

simplifies to

x = 4

. Only when

x = 4

is the original equation true.

No Solution: The equation simplifies to a false statement where the variables cancel but the remaining numbers don't match, like

5 = 7

0 = 3

. This tells you no value of

x

will ever make the original equation true. For instance,

2x + 6 = 2x + 5

simplifies to

6 = 5

, which is impossible.

Infinitely Many Solutions: The equation simplifies to a true statement where the variables cancel and the numbers do match, like

6 = 6

0 = 0

. This means every value of

x

satisfies the equation. For example,

2x + 6 = 2(x + 3)

simplifies to

6 = 6

after distributing and combining like terms. The equation is an identity—it's true for all

x

Why Variables Cancel: Recognizing Identity and Contradiction

Understanding why variables cancel is key to understanding these three cases.

When both sides of an equation contain the variable, and you combine like terms or simplify, you're revealing whether the two sides were really the same expression all along.

Consider

2x + 6 = 2(x + 3)

. On the right side, distribute to get

2x + 6 = 2x + 6

. Both sides are identical. Subtract

2x

from both sides:

6 = 6

. This is always true. The equation is an identity: it says the two sides are literally the same expression written different ways.

Now consider

2x + 6 = 2x + 5

. Subtract

2x

from both sides:

6 = 5

. This is always false, no matter what

x

is. The equation is a contradiction: the two sides can never be equal because after removing the variable part, what remains is impossible.

When the variable disappears and you're left with a true statement (like

6 = 6

), infinitely many values of

x

work. When it disappears and you're left with a false statement (like

6 = 5

), no values work. And when the variable doesn't completely disappear, you get exactly one value.

Solving to Determine the Number of Solutions

The process is the same as solving any linear equation: use inverse operations to simplify both sides and isolate the variable. What's new is that you must pay attention to what you end up with.

Step 1: Simplify each side (distribute, combine like terms).

Step 2: Move variable terms to one side and constants to the other (or move all terms to one side).

Step 3: Look at what remains:

Result	Number of Solutions	Meaning
$x = a$ (a number)	One	Exactly one value works
$a = a$ (both sides identical numbers)	Infinitely many	All real numbers work
$a = b$ where $a \ne b$	None	No number works

For example, solve

3(x + 2) = 3x + 6

:

Distribute:

3x + 6 = 3x + 6

Subtract

3x

6 = 6

✓ Infinitely many solutions.

For

3(x + 2) = 3x + 7

:

Distribute:

3x + 6 = 3x + 7

Subtract

3x

6 = 7

✗ No solution.

Common Mistakes and Where Students Go Wrong

Many students assume every equation has exactly one solution, or they stop solving too early.

Mistake 1: Forgetting to distribute or combine like terms fully. If you skip these steps, you might miss that the variable disappears. Always simplify both sides completely before moving terms around.

Mistake 2: Stopping work when a variable term cancels. When you subtract

2x

from both sides and the

x

terms disappear, do not stop. Keep going and evaluate what remains. A statement like

5 = 5

5 = 3

tells you the answer.

Mistake 3: Confusing "no solution" with "the solution is zero." If the equation simplifies to

x = 0

, that is one solution (and it is correct). If it simplifies to

3 = 5

, there is no solution at all—not even zero works.

Mistake 4: Thinking that if the variable cancels, something is wrong with your work. It is not wrong; it is a feature. It reveals that the equation is either always true (identity) or always false (contradiction).

Key terms

Identity.: An equation that is true for all values of the variable. When solved, it simplifies to a true statement like $0 = 0$ or $6 = 6$ .
Contradiction.: An equation that is false for all values of the variable. When solved, it simplifies to a false statement like $0 = 5$ or $3 = 7$ .
One solution.: An equation that is true for exactly one value of the variable, which you find by solving to get $x = a$ (where $a$ is a specific number).
No solution.: An equation that is never true, no matter what value you substitute for the variable. It simplifies to a false statement.
Infinitely many solutions.: An equation that is always true. Every real number satisfies it. It simplifies to an identity like $0 = 0$ .
Inverse operations.: Operations that undo each other, such as addition and subtraction, or multiplication and division. Used to isolate the variable when solving.

Worked example

Determine whether the equation

4x - 8 = 4(x - 2)

has one solution, no solution, or infinitely many solutions. Show your work.

Step 1: Distribute on the right side.

4x - 8 = 4(x - 2)

4x - 8 = 4x - 8

Step 2: Subtract $4x$ from both sides.

4x - 8 - 4x = 4x - 8 - 4x

-8 = -8

Step 3: Interpret the result.

We get the statement

-8 = -8

, which is always true. Both sides of the original equation are identical expressions.

Answer: Infinitely many solutions.

Every real number is a solution to this equation. We can verify: try

x = 0

4(0) - 8 = 4(0 - 2) \Rightarrow -8 = -8

✓. Try

x = 5

4(5) - 8 = 4(5 - 2) \Rightarrow 20 - 8 = 4(3) \Rightarrow 12 = 12

✓. The equation is an identity.

Practice questions

Solve

5x + 10 = 5(x + 2)

and determine the number of solutions.

Answer: Infinitely many solutions

Distribute on the right:

5x + 10 = 5x + 10

. Subtract

5x

from both sides:

10 = 10

. This is a true statement, so the equation is an identity. Every real number is a solution.

Which of the following equations has no solution?

$2x + 3 = 2x + 3$
$2x + 3 = 2x + 5$
$2x + 3 = 5$
$2x + 5 = 2x + 3 + 2$

Answer: $2x + 3 = 2x + 5$

Simplify each choice. The first simplifies to

0 = 0

(infinitely many solutions). The second choice simplifies to

3 = 5

after subtracting

2x

from both sides—a false statement, so no solution. The third simplifies to

x = 1

(one solution). The fourth simplifies to

2x + 5 = 2x + 5

, or

0 = 0

(infinitely many solutions).

Solve

3(x - 1) = 3x

and explain what this tells you about whether the equation has a solution.

Answer: No solution

Distribute:

3x - 3 = 3x

. Subtract

3x

from both sides:

-3 = 0

. This is a false statement that does not depend on

x

. No value of

x

can make

-3 = 0

true, so the equation has no solution. The two sides of the original equation can never be equal.

FAQ

If I get $x = 0$ when I solve, does that mean no solution?: No. If your answer is $x = 0$ , that is one solution. Zero is a real number, and it is the value that makes the equation true. 'No solution' means that no value at all works—you get that when the equation simplifies to a false statement like $5 = 3$ or $0 = 2$ .
How can I tell the difference between 'no solution' and 'infinitely many solutions'?: After you simplify and the variable cancels, look at what remains. If you get a true statement (like $0 = 0$ or $5 = 5$ ), you have infinitely many solutions. If you get a false statement (like $0 = 3$ or $5 = 2$ ), you have no solution. The variable cancels in both cases, but the number statement that's left tells you which one it is.
Why does the variable sometimes disappear when I solve?: The variable disappears when the expression on both sides of the equation is actually the same underneath, or fundamentally incompatible. If they're the same (an identity like $2x + 6 = 2(x + 3)$ ), then the equation is always true. If they're incompatible (a contradiction like $2x + 6 = 2x + 5$ ), then the equation is never true. When you solve, you're revealing which one it is.
Can I check my answer by substituting a number if the equation has infinitely many solutions?: Yes. If the equation has infinitely many solutions, you can pick any real number and substitute it in, and it will make both sides equal. For example, in $2x + 6 = 2(x + 3)$ , try $x = 10$ : left side is $2(10) + 6 = 26$ , right side is $2(10 + 3) = 26$ . It works. Try $x = -5$ : left side is $2(-5) + 6 = -4$ , right side is $2(-5 + 3) = -4$ . It works again. This confirms infinitely many solutions.

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