Solving Equations — Print
Undo what was done, in reverse order, doing the same thing to both sides every time.
What the practice questions cover
These questions use 2 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Distribute — Multiply a bracket out: 3(x + 4) = 3x + 12.
- Parallel — Same slope, different intercept.
Think of a number. Multiply it by 3. Add 4. Tell me the answer — 19.
I can now tell you the number you started with, and I never saw it.
Work the chain backwards. The last thing done was add 4, so undo it first: 19 - 4 = 15.
Before that it was multiplied by 3, so undo that next: 15 ÷ 3 = 5. Your number was 5.
Notice the order. Forwards it was × 3 then +4; backwards it is -4 then ÷ 3 — reversed, not repeated.
Written as algebra that chain is 3x + 4 = 19, and solving it is exactly the unwinding you just did. The equation is a record of what was done; solving is doing it in reverse.
Reverse the order of operations
To build 3x + 5 from x, you multiply by 3, then add 5. To take it apart, go backwards: subtract 5 first, then divide by 3.
3x + 5 = 20 ⇒ 3x = 15 ⇒ x = 5.
Undoing in the wrong order still sometimes works, but it usually creates fractions you did not need.
The order to undo
Clear + and - first. Then clear × and ÷. The opposite of PEMDAS.
Variables on both sides
Collect the variable terms on one side and the numbers on the other.
5x - 2 = 3x + 8. Subtract 3x from both sides: 2x - 2 = 8. Add 2: 2x = 10. Divide: x = 5.
A small tip: move the variable to whichever side keeps the coefficient positive. It prevents sign errors.
Equations with parentheses and fractions
Distribute first: 2(x + 3) = 14 becomes 2x + 6 = 14, so x = 4.
Fractions: multiply every term by the common denominator to clear them. x3 + 12 = 2 times 6 gives 2x + 3 = 12, so x = 4.5.
Multiply every term, including the ones without fractions.
One line, one change
Write each step on its own line and change only one thing per line. It is slower on the page and much faster overall, because you can find your mistake instead of starting over.
Always check
Substitute your answer back into the original equation. Not into a line you wrote — into the original, in case you made an error early.
x = 5 in 5x - 2 = 3x + 8: left side = 23, right side = 23. ✓
No solution, or every solution
Sometimes the variable vanishes. 2x + 3 = 2x + 7 becomes 3 = 7 — false, so there is no solution.
2x + 6 = 2(x + 3) becomes 6 = 6 — always true, so every number works.
These are not mistakes. They are real answers.
Worked examples
Variables both sides
Solve 7x - 4 = 2x + 21.
- Subtract 2x from both sides: 5x - 4 = 21 Gather the variables on the left, where the coefficient is bigger.
- Add 4 to both sides: 5x = 25 Clear the constant.
- Divide by 5: x = 5 Undo the multiplication.
- Check: 7(5) - 4 = 31 and 2(5) + 21 = 31 ✓ Both sides agree, so it is right.
x = 5
Clear the fractions
Solve 2x3 - 1 = x2 + 4.
- Common denominator of 3 and 2 is 6. Multiply everything by 6 to clear fractions.
- 4x - 6 = 3x + 24 6 · 2x3 = 4x, 6 · 1 = 6, 6 · x2 = 3x, 6 · 4 = 24.
- Subtract 3x: x - 6 = 24 Variables to one side.
- Add 6: x = 30 Check in the original: 603 - 1 = 19 and 302 + 4 = 19 ✓
x = 30
A word problem end to end
A taxi charges $3.50 plus $2.25 per mile. A ride cost $23.75. How many miles?
- Let m = miles. Name the unknown.
- 3.50 + 2.25m = 23.75 Flat fee plus per-mile charge equals the total.
- 2.25m = 20.25 Subtract the flat fee from both sides.
- m = 20.25 ÷ 2.25 = 9 Divide by the rate.
9 miles.
Words to know
- Inverse operation
- The operation that undoes another: + undoes -, × undoes ÷.
- Isolate
- Get the variable alone on one side.
- Two-step equation
- One needing two inverse operations, like 3x + 5 = 20.
- Variables on both sides
- An equation like 5x - 2 = 3x + 8.
Mistakes to watch for
Doing something to one side only. The equals sign is a promise that both sides are the same amount; change one and the promise breaks, so every line after it is about a different problem.
Multiplying only some terms when clearing fractions. Multiplying by 12 must hit every term, including whole numbers that were never fractions. Miss one and the equation quietly changes.
Checking against a rewritten line instead of the original. If you made an error on line two, checking against line three will happily confirm it. Always substitute back into the equation you were given.
Undoing the operations in the original order. If the chain was × 3 then +4, the undo is -4 then ÷ 3. Reversing the order is the whole point, and doing them forwards gives a wrong answer that looks tidy.
"Think of a number" puzzles are ancient. The Rhind Papyrus from about 1650 BC contains one: a quantity and its seventh, added together, make 19.
The Egyptian scribe solved it by guessing 7, seeing that it gave 8 instead of 19, and then scaling the guess. The method is called false position.
It works, it is clever, and it is exactly what algebra was invented to replace.
Problem 1
Solve x + 7 = 19.
Worked answer
12
x = 19 - 7 = 12
Problem 2
Solve 5x = 45.
Worked answer
9
x = 45 ÷ 5 = 9
Problem 3
Solve x4 = 7.
Worked answer
28
x = 7 × 4 = 28
Problem 4
Solve 3x + 2 = 17.
Worked answer
5
3x = 15
x = 5
Problem 5
Solve x - 8 = -3.
Worked answer
5
x = -3 + 8 = 5
Problem 6
Solve 4x - 9 = 23.
Worked answer
8
4x = 32
x = 8
Problem 7
Solve 7x - 4 = 2x + 21.
Worked answer
5
5x - 4 = 21
5x = 25
x = 5
Check: 31 = 31 ✓
Problem 8
Solve 2(x + 3) = 14.
Worked answer
4
2x + 6 = 14
2x = 8
x = 4
Problem 9
Solve 5 - 2x = 17.
Worked answer
-6
-2x = 12
x = -6
Check: 5 - 2(-6) = 5 + 12 = 17 ✓
Problem 10
Solve 3(2x - 1) = 21.
Worked answer
4
2x - 1 = 7
2x = 8
x = 4
Problem 11
Solve x3 + 5 = 9.
Worked answer
12
x3 = 4
x = 12
Problem 12
Solve 6x + 8 = 4x - 10.
Worked answer
-9
2x + 8 = -10
2x = -18
x = -9
Problem 13
Solve 2x3 - 1 = x2 + 4.
Worked answer
30
× 6: 4x - 6 = 3x + 24
x - 6 = 24
x = 30
Check: 20 - 1 = 19 and 15 + 4 = 19 ✓
Problem 14
Solve 5(x - 2) = 3(x + 4).
Worked answer
11
5x - 10 = 3x + 12
2x - 10 = 12
2x = 22
x = 11
Problem 15
Solve 2x + 3 = 2x + 7.
- x = 2
- x = 4
- No solution
- Every number works
Worked answer
No solution
Subtract 2x: 3 = 7
That is false no matter what x is
So there is no solution. The two sides are parallel lines that never meet.
Problem 16
A taxi charges $3.50 plus $2.25 per mile. A ride cost $23.75. How many miles?
Worked answer
9
2.25m = 23.75 - 3.50 = 20.25
m = 20.25 ÷ 2.25 = 9 miles