Equations with the letter on both sides

Some­times the let­ter appears on both sides of an equa­tion, as in $x - 2 = 8 - x$. You can­not find $x$ until all the $x$ terms are together on one side. In this les­son you learn to gather the $x$ terms on the left and the plain num­bers on the right, using the same trans­pos­ing rule as before.

Gather the terms#

Use the rule you already know: when a term moves to the other side, its sign changes. This works for $x$ terms as well as for num­bers.

1. Move all the $x$ terms to the left side.
2. Move all the plain num­bers to the right side.
3. Add the like terms on each side.
4. Divide by the num­ber in front of $x$.

You will need the facts for adding like terms: $x + x = 2x$, $3x + 2x = 5x$ and $10x - 8x = 2x$.

Exam­ple 1#

Solve $x - 2 = 8 - x$.

The $-2$ moves right and becomes $+2$. The $-x$ moves left and becomes $+x$.

$$\displaystyle \begin{aligned}x - 2 &= 8 - x \\ \Rightarrow x + x &= 8 + 2 \\ \Rightarrow 2x &= 10 \\ \Rightarrow x &= \frac{10}{2} \\ \Rightarrow x &= 5\end{aligned}$$

Check. Left side: $5 - 2 = 3$. Right side: $8 - 5 = 3$. Both sides are 3.

Exam­ple 2#

Solve $3x - 10 = 15 - 2x$.

The $-10$ moves right as $+10$. The $-2x$ moves left as $+2x$.

$$\displaystyle \begin{aligned}3x - 10 &= 15 - 2x \\ \Rightarrow 3x + 2x &= 15 + 10 \\ \Rightarrow 5x &= 25 \\ \Rightarrow x &= \frac{25}{5} \\ \Rightarrow x &= 5\end{aligned}$$

Check: left side $15 - 10 = 5$, right side $15 - 10 = 5$.

Join like terms first#

If one side already has two $x$ terms, join them before you move any­thing.

Exam­ple 3#

Solve $7x - 3 + 3x = 11 + 8x$.

On the left, $7x$ and $3x$ are like terms: $7x + 3x = 10x$.

$$\begin{aligned}7x - 3 + 3x &= 11 + 8x \\ \Rightarrow 10x - 3 &= 11 + 8x \\ \Rightarrow 10x - 8x &= 11 + 3 \\ \Rightarrow 2x &= 14 \\ \Rightarrow x &= 7\end{aligned}$$

Check: left side $70 - 3 = 67$, right side $11 + 56 = 67$.

When the x terms leave a neg­a­tive#

Exam­ple 4#

Solve $8x + 123 = -256 + 9x$.

The 123 moves right as $-123$. The $9x$ moves left as $-9x$.

$$\displaystyle \begin{aligned}8x + 123 &= -256 + 9x \\ \Rightarrow 8x - 9x &= -256 - 123 \\ \Rightarrow -1x &= -379 \\ \Rightarrow x &= \frac{-379}{-1} \\ \Rightarrow x &= 379\end{aligned}$$

Here $8 - 9 = -1$, so the left side is $-1x$, which is $-x$. Divid­ing by $-1$ changes the sign: a neg­a­tive divided by a neg­a­tive is pos­i­tive. So $x = 379$, not $-379$.

Check: left side $8 \times 379 + 123 = 3032 + 123 = 3155$. Right side $-256 + 9 \times 379 = -256 + 3411 = 3155$.

Exam­ple 5#

Solve $5x - 7 = -9 + 3x$.

$$\begin{aligned}5x - 7 &= -9 + 3x \\ \Rightarrow 5x - 3x &= -9 + 7 \\ \Rightarrow 2x &= -2 \\ \Rightarrow x &= -1\end{aligned}$$

Check: left side $-5 - 7 = -12$, right side $-9 - 3 = -12$.

Equa­tions with a frac­tion#

Exam­ple 6#

Solve $\displaystyle \frac{3x}{2} + 9 = -9$.

$$\displaystyle \begin{aligned}\frac{3x}{2} + 9 &= -9 \\ \Rightarrow \frac{3x}{2} &= -9 - 9 \\ \Rightarrow \frac{3x}{2} &= -18 \\ \Rightarrow 3x &= -36 \\ \Rightarrow x &= -12\end{aligned}$$

Exam­ple 7#

Solve $-13x + 7 - 14x = -47$.

Join the $x$ terms first: $-13x - 14x = -27x$, because both signs are minus.

$$\displaystyle \begin{aligned}-13x + 7 - 14x &= -47 \\ \Rightarrow -27x + 7 &= -47 \\ \Rightarrow -27x &= -47 - 7 \\ \Rightarrow -27x &= -54 \\ \Rightarrow x &= \frac{-54}{-27} \\ \Rightarrow x &= 2\end{aligned}$$

Check: $-26 + 7 - 28 = -47$.

Remem­ber. $x$ terms to the left, num­bers to the right, and change the sign of any­thing that crosses the equals sign.

Com­mon mis­takes#

• Mov­ing a term with­out chang­ing its sign. In $3x - 10 = 15 - 2x$, the $-2x$ becomes $+2x$ on the left.
• For­get­ting to join like terms first. In $7x - 3 + 3x$, join $7x + 3x = 10x$ before mov­ing.
• Ignor­ing the sign of $-x$. If you reach $-x = -379$, then $x = 379$.
• Not check­ing both sides. Work out the left side and the right side sep­a­rately.

Key terms#

• Gather: bring all the like terms to the same side of the equa­tion.
• Like terms: terms with the same let­ter part, such as $3x$ and $2x$.

Prac­tice#

1. Solve $4x + 5 = x + 17$.
2. Solve $6x - 1 = 2x + 11$.
3. Solve $9 - x = x + 1$.
4. Solve $2x + 3 = 5x - 12$.
5. Solve $7x - 4 = 3x - 20$.
6. Solve $10 - 3x = 2x - 15$.
7. Solve $5x - 7 = -9 + 3x$.
8. Solve $4x + 2 = 18 + 2x$.
9. Solve $x + 40 = 3x$.
10. Solve $\displaystyle \frac{3x}{2} + 9 = -9$.
11. Solve $-13x + 7 - 14x = -47$.
12. Seven times a num­ber is 24 more than the num­ber. Find the num­ber.

Answers#