Unwrap the unknown
Undo what was done to the letter-number, one layer at a time.
Two things happened to x
In 5x − 4 = 7, the letter-number x was first multiplied by 5, and then 4 was taken away. We want x alone.
Which should we undo first?
What this lesson covers
The idea
To solve an equation systematically, remove a term added to or subtracted from the unknown term by applying the inverse operation to both sides, then remove the unknown's factor by dividing both sides by it.
Two things happened to x
In 5x − 4 = 7, the letter-number *x* was first multiplied by 5, and then 4 was taken away. We want *x* alone.
Which should we undo first?
- Undo the “− 4”
- Undo the “× 5”
- It makes no difference
Make x stand alone
Pick an operation and a number. It is done to both sides and written out. Keep going until the letter-number is alone on one side.
Remove the term, then remove the factor
The letter-number is wrapped in two layers. The outer one, the term added or subtracted (− 4), goes first. We undo it with the inverse operation on both sides. Then the factor (5) goes: divide both sides by it.
Check with the first line: 5 × 11/5 − 4 = 11 − 4 = 7.
To solve an equation step by step, remove the term added to or subtracted from the unknown term by the inverse operation on both sides, then remove the unknown’s factor by dividing both sides by it.
- Step | Equation
- start | 5x − 4 = 7
- add 4 to both sides | 5x − 4 + 4 = 7 + 4, so 5x = 11
- divide both sides by 5 | 5x ÷ 5 = 11 ÷ 5, so *x* = 11/5
Notes
Remove the term (add or subtract) with the inverse operation on both sides, then remove the factor by dividing both sides by it: 5x − 4 = 7 → 5x = 11 → *x* = 11/5.
Check yourself
Solve 2x + 3 = 15.
Answer: 6
2x + 3 − 3 = 15 − 3 gives 2x = 12. Then x = 12 ÷ 2 = 6.
To solve 7m − 9 = 40, what is the best first step?
Solve 11y − 5 = 61.
Answer: 6
11y − 5 + 5 = 61 + 5 gives 11y = 66. Then y = 66 ÷ 11 = 6.
Solve 4p + 20 = 0.
Answer: -5
4p + 20 − 20 = 0 − 20 gives 4p = −20. Then p = −20 ÷ 4 = −5. Check: 4 × (−5) + 20 = 0.
- Add 9 to both sides: 7m = 49. — correct. Yes! Adding 9 undoes the − 9. Then divide by 7.
- Subtract 9 from both sides: 7m − 18 = 31.. The term is − 9, so its inverse is + 9. Subtracting 9 makes it − 18.
- Divide both sides by 7: m − 9 = 40/7.. Dividing the whole LHS by 7 also divides the 9, so the LHS is m − 9/7, not m − 9. Remove the −9 first.