HSPT: Equations: solve in two steps
Learn to solve equations like ax + b = c by undoing each step on both sides, and to check your answer by substituting it back.
How do you solve an equation where x is multiplied and then has a number added to it?
Short answer
Undo the addition or subtraction first, then the multiplication or division, doing the same thing to both sides. Then check your answer in the equation.
Know first: Inverse operations, Integer arithmetic
Step 1 of 5
Undo the equation one step at a time
An equation like 5x + 15 = 60 is a balance: both sides are equal. To find x, undo what was done to it, and do the same thing to both sides so they stay equal.
Undo in reverse order. In 5x + 15, x was multiplied by 5 and then 15 was added. So subtract 15 first, then divide by 5.
| What was done to x | Undo it by | Do it to |
|---|---|---|
| Added a number | Subtracting it | Both sides |
| Subtracted a number | Adding it | Both sides |
| Multiplied by a number | Dividing by it | Both sides |
| Divided by a number | Multiplying by it | Both sides |
Step 2 of 5
Subtract, then divide
Worked example: A two-step equation
Solve for x: 5x + 15 = 60.
- A9
- B45
- C15
- D−3
- 1
Undo the addition
Subtract 15 from both sides: 5x = 45.
- 2
Undo the multiplication
Divide both sides by 5: x = 9.
- 3
Check
5(9) + 15 = 45 + 15 = 60.
- 4
See why the other choices are there
45 stops before dividing. 15 adds 15 instead of subtracting: (60 + 15) ÷ 5. −3 divides only the 60 by 5 and then subtracts 15.
Answer
x = 9
Find the wrong step
A student solves 2x − 8 = 14.
- 1
Subtracts 8 from both sides: 2x = 6.
What went wrong
The equation subtracts 8 from 2x, so the way to undo it is to add 8.
- 2
Divides by 2: x = 3.
- 3
Answers 3.
The fix
Add 8: 2x = 22, so x = 11. Check: 2(11) − 8 = 14.
Step 3 of 5
Solve and check
Check yourself · Question 1
Solve for x: 2x + 6 = 38.
Answer: C
Subtract 6 from both sides: 2x = 32. Divide by 2: x = 16. Check: 2(16) + 6 = 38.
Trap answer: 32
Stopping one step early
- Why it looks right
- After subtracting 6, the number 32 is sitting alone on one side, so it feels like the answer.
- Why it's wrong
- 32 is the value of 2x, not x. One more step, dividing by 2, finishes the job.
- The right answer
- 16, because 2(16) + 6 = 38.
Check yourself · Question 2
Solve for x: x/5 + 2 = 9.
Answer: C
Subtract 2 from both sides: x/5 = 7. Multiply both sides by 5: x = 35. Check: 35 ÷ 5 + 2 = 7 + 2 = 9.
Common mistake
"I see the number next to x and divide by it first."
- Why it's tempting
- Getting x alone is the goal, and the number touching x looks like the obvious first target.
- Do this instead
- Clear the added or subtracted number first. Then divide or multiply. If you divide first, you must divide every term, which is easy to get wrong.
Step 4 of 5
What to remember
Remember
- Do the same thing to both sides so the equation stays balanced.
- Undo adding or subtracting first, then multiplying or dividing.
- Check by putting your answer back into the equation.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
Which step did you undo first in the practice question, and how did you check your answer?