Equations with adding and taking away

"I am think­ing of a num­ber. When I add 2 to it, I get 8. What is my num­ber?" You can prob­a­bly answer at once: 6. In alge­bra this puz­zle is writ­ten as $x + 2 = 8$, and find­ing $x$ is called solv­ing the equa­tion. In this les­son you learn a sure method for equa­tions like this, where a num­ber is added to $x$ or taken away from it, includ­ing equa­tions whose answers are neg­a­tive inte­gers.

What an equa­tion is#

An equa­tion is a state­ment that two amounts are equal. It has an equals sign, a left side and a right side.

$$x + 2 = 8$$

Here the left side is $x + 2$ and the right side is 8. The let­ter $x$ stands for a num­ber we do not know yet.

To solve an equa­tion means to find the value of $x$ that makes both sides equal. That value is called the solu­tion.

Think of a bal­ance with two pans. The equals sign says the two pans weigh the same. If you take the same amount off both pans, they still bal­ance.

Equa­tions like x + a = b#

Exam­ple 1#

Solve $x + 2 = 8$.

Two has been added to $x$. To get $x$ on its own, take 2 away from both sides. The 2 dis­ap­pears from the left, and 8 becomes $8 - 2$ on the right.

$$\begin{aligned}x + 2 &= 8 \\ \Rightarrow x &= 8 - 2 \\ \Rightarrow x &= 6\end{aligned}$$

The arrow $\Rightarrow$ is read "implies". It means "so the next line is true".

Check. Put 6 in place of $x$: $6 + 2 = 8$. Both sides are 8, so $x = 6$ is right.

Look at what hap­pened to the 2. It was added on the left. It moved to the right side and became taken away. This is called trans­pos­ing.

Remem­ber. When a num­ber moves to the other side of the equals sign, its sign changes. Plus becomes minus, and minus becomes plus.

Exam­ple 2#

Solve $x + 5 = 11$.

$$\begin{aligned}x + 5 &= 11 \\ \Rightarrow x &= 11 - 5 \\ \Rightarrow x &= 6\end{aligned}$$

Check: $6 + 5 = 11$.

Exam­ple 3#

Solve $x + 7 = -8$.

The $+7$ moves across and becomes $-7$.

$$\begin{aligned}x + 7 &= -8 \\ \Rightarrow x &= -8 - 7 \\ \Rightarrow x &= -15\end{aligned}$$

To work out $-8 - 7$, use the rule for adding inte­gers. Both signs are minus, so add $8 + 7 = 15$ and keep the minus sign.

Check: $-15 + 7 = -8$. The signs dif­fer, the dif­fer­ence is 8, and the big­ger num­ber, 15, is neg­a­tive. So both sides are $-8$.

Equa­tions like x − a = b#

Now a num­ber is taken away from $x$. To undo tak­ing away, add.

Exam­ple 4#

Solve $x - 2 = 4$.

The $-2$ moves across and becomes $+2$.

$$\begin{aligned}x - 2 &= 4 \\ \Rightarrow x &= 4 + 2 \\ \Rightarrow x &= 6\end{aligned}$$

Check: $6 - 2 = 4$.

Exam­ple 5#

Solve $x - 8 = -15$.

$$\begin{aligned}x - 8 &= -15 \\ \Rightarrow x &= -15 + 8 \\ \Rightarrow x &= -7\end{aligned}$$

For $-15 + 8$ the signs are dif­fer­ent. The dif­fer­ence is $15 - 8 = 7$, and the big­ger num­ber, 15, is neg­a­tive. So the answer is $-7$.

Check: $-7 - 8 = -15$.

Exam­ple 6#

Solve $x - 8 = -2$.

$$\begin{aligned}x - 8 &= -2 \\ \Rightarrow x &= -2 + 8 \\ \Rightarrow x &= 6\end{aligned}$$

This time the big­ger num­ber, 8, is pos­i­tive, so the answer is pos­i­tive. Check: $6 - 8 = -2$.

Exam­ple 7#

Solve $x - 15 = -27$.

$$\begin{aligned}x - 15 &= -27 \\ \Rightarrow x &= -27 + 15 \\ \Rightarrow x &= -12\end{aligned}$$

The dif­fer­ence is $27 - 15 = 12$, and 27 is the big­ger num­ber with a minus sign, so $x = -12$. Check: $-12 - 15 = -27$.

The inte­ger facts you need#

Every step above used the rules for adding inte­gers. Here they are again with small num­bers.

The two rules so far#

Remem­ber. Always check your answer by putting it back into the equa­tion. If both sides come out equal, the answer is right.

Com­mon mis­takes#

• For­get­ting to change the sign. In $x + 7 = -8$, the 7 moves across as $-7$, not $+7$.
• Get­ting the inte­ger arith­metic wrong. $-8 - 7$ is $-15$, not $-1$.
• Not check­ing. A quick check catches most slips.
• Writ­ing the answer as a sum. Fin­ish the arith­metic: write $x = 6$, not $x = 8 - 2$.

Key terms#

• Equa­tion: a state­ment that two amounts are equal, such as $x + 2 = 8$.
• Solve: find the value of the let­ter that makes the equa­tion true.
• Solu­tion: the value that makes the equa­tion true.
• Trans­pose: move a term to the other side of the equals sign, chang­ing its sign.

Prac­tice#

1. Solve $x + 9 = 15$.
2. Solve $x + 125 = 256$.
3. Solve $x + 12 = -28$.
4. Solve $x + 3 = -3$.
5. Solve $x - 11 = 7$.
6. Solve $x - 19 = 32$.
7. Solve $x - 27 = -45$.
8. Solve $x - 25 = -10$.
9. Solve $x - 6 = -6$.
10. Solve $x + 14 = 9$.
11. When 8 is added to a num­ber, the result is $-3$. Find the num­ber.
12. Check whether $x = -4$ is the solu­tion of $x - 5 = -9$.

Answers#