Solving for the unknown (x)
When we "solve for the unknown," we are finding the value of a letter (usually x) that makes a math sentence true. Think of an equation like a balance scale: whatever you do to one side, you must do to the other side so it stays even.
Basic idea (the balancing rule)
If you have an equation, do the same thing to both sides to keep it balanced. Use opposite (inverse) operations to get x by itself.
Inverse operations
- Addition ⇄ Subtraction (undo + by -)
- Multiplication ⇄ Division (undo × by ÷)
Step-by-step method
- Look at the equation and see what operation is done to x.
- Use the inverse operation to remove that from x.
- Do the same thing to the other side of the = sign.
- Keep going until x is alone on one side. Then check your answer by plugging it back in.
Examples
1) x + 5 = 12
- Operation on x: +5. Inverse: -5.
- Subtract 5 from both sides: x + 5 - 5 = 12 - 5
- So x = 7.
- Check: 7 + 5 = 12 ✔
2) x - 4 = 9
- Operation on x: -4. Inverse: +4.
- Add 4 to both sides: x - 4 + 4 = 9 + 4
- So x = 13.
- Check: 13 - 4 = 9 ✔
3) 3x = 15
- Operation on x: multiplied by 3. Inverse: divide by 3.
- Divide both sides by 3: (3x)/3 = 15/3
- So x = 5.
- Check: 3×5 = 15 ✔
4) x / 4 = 6
- Operation on x: divided by 4. Inverse: multiply by 4.
- Multiply both sides by 4: (x/4)×4 = 6×4
- So x = 24.
- Check: 24 / 4 = 6 ✔
5) Two-step 2x + 3 = 11
- First remove +3 by subtracting 3 from both sides: 2x = 11 - 3 = 8.
- Now remove ×2 by dividing both sides by 2: x = 8 ÷ 2 = 4.
- Check: 2×4 + 3 = 8 + 3 = 11 ✔
Quick tips
- Always do the same thing to both sides.
- Work step by step: remove addition/subtraction first, then multiplication/division (for simple two-step problems).
- After you find x, always substitute it back into the original equation to check.
Practice problems (try them)
1) x + 8 = 14
2) 5x = 20
3) x/3 = 7
4) 4x - 2 = 10
Answers
1) x = 6
2) x = 4
3) x = 21
4) 4x - 2 = 10 → add 2: 4x = 12 → divide 4: x = 3
Good job! Keep practicing and remember the balancing rule: do the same to both sides.