Math Core

Gym A charges $40 to join plus $5 per visit. Gym B charges $20 to join plus $9 per visit. After how many visits do they cost the same? Setting the costs equal gives $40 + 5v = 20 + 9v$ , and now the variable appears on both sides. You need one new move to handle it.

Collect the variable terms

The properties of equality let you add or subtract any number from both sides. A variable term like $5v$ is also a number (you just don't know its value yet), so you can subtract it from both sides too. That moves all the variable terms to one side.

Variables on both sides

Simplify each side: distribute and combine like terms.
Add or subtract a variable term so the variable appears on only one side.
Add or subtract a constant so the numbers are on the other side.
Divide (or multiply) to isolate the variable, then check.

Back to the gyms. Subtract $5v$ from both sides, so the variable is only on the right:

\begin{aligned} 40 + 5v &= 20 + 9v \\ 40 &= 20 + 4v && \text{subtract } 5v \\ 20 &= 4v && \text{subtract } 20 \\ 5 &= v && \text{divide by } 4 \end{aligned}

After $5$ visits both gyms cost the same: $40 + 5(5) = 65$ dollars and $20 + 9(5) = 65$ dollars. ✓

Tip

You may collect the variable on either side, and you'll get the same answer. A handy habit is to subtract the smaller variable term. Here subtracting $5v$ left $4v$ , a positive coefficient, which means one fewer negative sign to track.

Worked example: Picturing the solution

Solve $5x - 4 = 3x + 6$ .

\begin{aligned} 5x - 4 &= 3x + 6 \\ 2x - 4 &= 6 && \text{subtract } 3x \\ 2x &= 10 && \text{add } 4 \\ x &= 5 && \text{divide by } 2 \end{aligned}

Check: $5(5) - 4 = 21$ and $3(5) + 6 = 21$ . ✓

You can also see this solution on a graph. Treat each side as a line: $y = 5x - 4$ and $y = 3x + 6$ . The solution is the $x$ -value where the two sides are equal, which is where the lines cross.

The two sides are equal where the lines meet, at x = 5.Open in grapher →

Worked example: Negative coefficients

Solve $7 - 2n = 4n + 25$ .

The smaller variable term is $-2n$ , so add $2n$ to both sides.

\begin{aligned} 7 - 2n &= 4n + 25 \\ 7 &= 6n + 25 && \text{add } 2n \\ -18 &= 6n && \text{subtract } 25 \\ -3 &= n && \text{divide by } 6 \end{aligned}

Check: $7 - 2(-3) = 13$ and $4(-3) + 25 = 13$ . ✓

Worked example: Parentheses on both sides

Solve $3(y + 4) = 5(y - 2) + 6$ .

Simplify each side first, then collect.

\begin{aligned} 3(y + 4) &= 5(y - 2) + 6 \\ 3y + 12 &= 5y - 10 + 6 && \text{distribute} \\ 3y + 12 &= 5y - 4 && \text{combine constants} \\ 12 &= 2y - 4 && \text{subtract } 3y \\ 16 &= 2y && \text{add } 4 \\ 8 &= y && \text{divide by } 2 \end{aligned}

Check: $3(12) = 36$ and $5(6) + 6 = 36$ . ✓

Common mistake

Move a term by doing the same operation to both sides, not by "hopping it over." Students who think of it as hopping often forget to change the sign: from $5x - 4 = 3x + 6$ they write $5x + 3x = 6 + 4$ . Subtracting $3x$ from both sides makes the sign change automatic and correct: $5x - 3x = 6 + 4$ .

No solution or infinitely many

Sometimes the variable terms cancel completely. What's left tells you what happened.

A false statement means no solution. Solve $2(x + 3) = 2x + 5$ :

\begin{aligned} 2x + 6 &= 2x + 5 \\ 6 &= 5 && \text{subtract } 2x \end{aligned}

$6 = 5$ is never true, no matter what $x$ is. The equation has no solution. (The left side is always exactly $1$ more than the right.)

A true statement means every number is a solution. Solve $4x - 8 = 4(x - 2)$ :

\begin{aligned} 4x - 8 &= 4x - 8 \\ -8 &= -8 && \text{subtract } 4x \end{aligned}

$-8 = -8$ is always true. The two sides are the same expression, so every real number is a solution. An equation like this is called an identity.

Definition

Identity

An identity is an equation that is true for every value of the variable. When solving leads to a true statement like $3 = 3$ , the equation is an identity and has infinitely many solutions.

Common mistake

Don't write " $x = 0$ " when the variables cancel. Getting $6 = 5$ means no value works; getting $-8 = -8$ means every value works. Neither result says the answer is $0$ .

Practice

Practice 1

Solve $9x - 2 = 4x + 18$ .

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

Practice 2

Solve $3a + 11 = 7a - 9$ .

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

Practice 3

Solve $-2m + 8 = 3m + 23$ .

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

Practice 4

Solve $6(k - 1) = 2(k + 7)$ .

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

Practice 5

How many solutions does $3(2x - 4) = 6x - 12$ have?

Practice 6

How many solutions does $5x + 2 - x = 4x - 7$ have?

Practice 7

Solve $\dfrac{x}{2} + 3 = \dfrac{x}{5} + 9$ .

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

Practice 8

Candle A is $30$ cm tall and burns down $2$ cm per hour. Candle B is $22$ cm tall and burns down $1$ cm per hour. Both are lit at the same time. After how many hours are they the same height?

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