Math Core

So far, every equation you've solved had exactly one answer. But try solving $2(x + 4) = 2x + 5$ and something strange happens: the variable disappears. That isn't a mistake. It's the equation telling you that it has no solution, or in other cases, infinitely many.

Three possible outcomes

When you solve a linear equation in one variable, you always end up in one of three places.

How many solutions?

One solution: you end with $x = a$ for a single number $a$ .
No solution: the variable cancels and you're left with a false statement, like $8 = 5$ .
Infinitely many solutions: the variable cancels and you're left with a true statement, like $6 = 6$ . Every number works.

Worked example: One solution

Solve $4x + 1 = 2x + 7$ .

\begin{aligned} 4x + 1 &= 2x + 7 \\ 2x + 1 &= 7 \\ 2x &= 6 \\ x &= 3 \end{aligned}

The equation is true only when $x = 3$ . It has one solution.

Worked example: No solution

Solve $2(x + 4) = 2x + 5$ .

\begin{aligned} 2(x + 4) &= 2x + 5 \\ 2x + 8 &= 2x + 5 \\ 8 &= 5 && \text{subtract } 2x \end{aligned}

The $x$ terms cancel and leave $8 = 5$ , which is never true. No matter what number you try, the left side is always $3$ more than the right side. The equation has no solution.

Worked example: Infinitely many solutions

Solve $3(x + 2) = 3x + 6$ .

\begin{aligned} 3(x + 2) &= 3x + 6 \\ 3x + 6 &= 3x + 6 \\ 6 &= 6 && \text{subtract } 3x \end{aligned}

The statement $6 = 6$ is always true. Both sides are the same expression written two ways, so every value of $x$ works. Try $x = 10$ : $3(12) = 36$ and $30 + 6 = 36$ . The equation has infinitely many solutions.

Common mistake

Don't write " $x = 0$ " when the variable cancels. Getting $0 = 0$ means every number is a solution, and getting something like $0 = 3$ means no number is. But if you end with $x = 0$ itself, that's one solution: the number zero.

A shortcut: compare the two sides

You can often tell how many solutions there are without solving all the way. First simplify each side into the form $ax + b$ (distribute and combine like terms). Then compare.

After simplifying…	Example	Number of solutions
Different $x$ coefficients	$4x + 1 = 2x + 7$	one
Same $x$ coefficient, different constants	$2x + 8 = 2x + 5$	none
Same $x$ coefficient, same constant	$3x + 6 = 3x + 6$	infinitely many

Why does this work? If the coefficients are different, subtracting one variable term leaves some $x$ behind, and you can solve for it. If the coefficients match, the $x$ terms cancel, and only the constants are left to compare.

Seeing it on a graph

You can picture each side of an equation as a line. The solution is where the two lines meet. For $4x + 1 = 2x + 7$ , the lines $y = 4x + 1$ and $y = 2x + 7$ cross at $x = 3$ .

The lines cross once, at x = 3: one solution.Open in grapher →

For $2x + 8 = 2x + 5$ , the lines $y = 2x + 8$ and $y = 2x + 5$ have the same steepness, so they run side by side and never meet.

The lines never meet: no solution.Open in grapher →

When both sides are the same expression, the two lines are the same line, so they meet everywhere.

Worked example: Build an equation

What number should go in the box so that $5x - 2 = 5x + \square$ has infinitely many solutions? What about no solution?

Both sides already have $5x$ . For infinitely many solutions, the constants must also match, so the box must be $-2$ . For no solution, the constants must be different, so any number other than $-2$ works, such as $4$ .

Tip

Before you decide, always simplify each side completely. In $3(x - 2) + x = 2(2x - 3) + x$ , the sides look alike, but they simplify to $4x - 6$ and $5x - 6$ . The coefficients differ, so there is one solution: $x = 0$ .

Practice

Practice 1

How many solutions does $6x + 4 = 6x - 1$ have?

Practice 2

How many solutions does $2(3x - 5) = 6x - 10$ have?

Practice 3

How many solutions does $5x - 3 = 2x + 9$ have?

Practice 4

How many solutions does $4(x + 1) - x = 3x + 4$ have?

Practice 5

How many solutions does $7x + 2 - 3x = 2(2x + 3)$ have?

Practice 6

How many solutions does $3(x - 2) + x = 2(2x - 3) + x$ have?

Practice 7

What value of $a$ makes $4(2x - 3) = ax - 12$ true for every value of $x$ (infinitely many solutions)?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

What value of $k$ makes $kx + 7 = 3x + 2$ have no solution?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.