The inverse crosses over
Remove a term from one side: its inverse appears on the other side.
2y + 7 = 21
If y = 7, then 2y + 7 = 21 is true: both sides are 21. Now we remove the + 7 from the left side. The left side becomes 2y.
What happens to the right side, 21, if the sides must still be equal?
What this lesson covers
The idea
When a term that is added or subtracted on one side of an equation is removed from that side, its additive inverse must appear as a term on the other side for the equality to hold.
2y + 7 = 21
If y = 7, then 2y + 7 = 21 is true: both sides are 21. Now we remove the + 7 from the left side. The left side becomes 2y.
What happens to the right side, 21, if the sides must still be equal?
- It becomes 21 − 7
- It becomes 21 + 7
- It stays 21
Remove it, then repair the other side
Tap the highlighted term to remove it. The sides stop agreeing. Then choose what the other side must become, and test your choice with the numbers.
The additive inverse appears
The term +7 was added. Removing it means subtracting 7. So the other side gets − 7. The term crosses to the other side as its additive inverse.
Why does it work? Removing +7 from the left is the same as subtracting 7 from both sides. The right side shows it as − 7.
When a term that is added or subtracted on one side of an equation is removed from that side, its additive inverse must appear as a term on the other side: 2y + 7 = 21 becomes 2y = 21 − 7.
- Equation | After removing the term
- 2y + 7 = 21 | 2y = 21 − 7
- 11y + (−5) = 61 | 11y = 61 − (−5)
- 6y + 7 = 4y + 21 | 6y + 7 − 4y = 21
- x − 9 = 4 | x = 4 + 9
Notes
A term removed from one side appears on the other side as its additive inverse: 2y + 7 = 21 becomes 2y = 21 − 7.
Check yourself
In 5x + 9 = 34, the term + 9 is removed from the left. What does the right side become?
Solve 2y − 6 = 10.
Answer: 8
2y = 10 + 6 = 16, so y = 16 ÷ 2 = 8. Check: 2 × 8 − 6 = 10.
The book shows a student who wrote 5z + 3z = −4 + 2 for 5z + 2 = 3z − 4. What went wrong?
Solve x + 12 = 5.
Answer: -7
x = 5 − 12 = −7. Check: −7 + 12 = 5.
- 34 − 9 — correct. Yes! The additive inverse of +9 is −9.
- 34 + 9. The term was +9. Its inverse is −9, not +9 again.
- 34. Something was removed on the left, so the right side must change too: 34 − 9.
- Nothing: 5z + 3z = −4 + 2 is correct.. It is not. With z = −3 the left is 5 × (−3) + 3 × (−3) = −24, but the right is −4 + 2 = −2.
- Both moved terms need their inverse: 5z − 3z = −4 − 2. — correct. Yes! 3z leaves the right side, so − 3z is added on the left. And +2 leaves the left, so − 2 appears on the right.
- Only 3z needs to change sign; the + 2 can stay.. The + 2 also left the left side. Its inverse − 2 must appear on the right: −4 − 2.