Learning Objectives
Use inverse operations to solve one-step equations.
Solve two-step equations while preserving equality.
Check a proposed solution in the original equation.
Prerequisites: expressions, signed arithmetic, and fractions. A solution is a value that makes an equation true. Solving is a process of finding such values, not simply moving symbols across an equals sign.
1. Keep Both Sides Equal
Imagine an equation as a balanced scale. Adding the same amount to both sides preserves the balance, as does subtracting the same amount. Multiplying or dividing both sides by the same nonzero number also preserves equality.
Operation on the variable | Inverse operation |
|---|---|
Add 6 | Subtract 6 |
Subtract 4 | Add 4 |
Multiply by 3 | Divide by 3 |
Divide by 5 | Multiply by 5 |
Worked Example: One Step
Solve x − 7 = 12. Add 7 to both sides: x = 19.
Check in the original equation: 19 − 7 = 12. The statement is true, so 19 is a solution.
2. Undo Operations in Reverse Order
In 3x + 5, the variable is first multiplied by 3, then 5 is added. To undo that process, subtract 5 first and then divide by 3. Writing a line for each step makes the reasoning visible.
Worked Example: Two Steps
Solve 3x + 5 = 23. Subtract 5 from both sides to obtain 3x = 18.
Divide both sides by 3: x = 6.
Check: 3(6) + 5 = 18 + 5 = 23. Both sides of the original equation agree.
3. A Negative Coefficient Still Uses Division
Worked Example: A Negative Multiplier
Solve 8 − 2x = 18. Subtract 8 from both sides: −2x = 10.
Divide both sides by −2 to get x = −5.
Check: 8 − 2(−5) = 8 + 10 = 18. A negative solution can be completely valid.
Do not divide by just 2 when the coefficient is −2 unless you keep the remaining negative sign and continue solving. The coefficient includes its sign.
4. Fractions Can Be Undone Too
Worked Example: A Divided Variable
Solve x/4 − 3 = 2. Add 3 to both sides: x/4 = 5.
Multiply both sides by 4: x = 20.
Check: 20/4 − 3 = 5 − 3 = 2. Multiplication undoes division by 4.
Not every solution is an integer. For example, 2x = 7 gives x = 7/2. Keep an exact fraction unless the question asks for a decimal or a specified rounding.
Common Mistakes
Apply an operation to both sides, not just the side containing the variable.
Undo the outside addition or subtraction before a multiplication when following the reverse-order method.
Check the original equation so an earlier algebra error does not get hidden.
5. Practice
1. Solve x + 9 = 4.
2. Solve −5x = 35.
3. Solve 4x − 7 = 13.
4. Solve x/3 + 2 = 8.
5. Is x = 3 a solution of 5x + 1 = 17? If not, find the solution.
Worked Solutions
1. Subtract 9: x = −5. Check: −5 + 9 = 4.
2. Divide by −5: x = −7. Check: (−5)(−7) = 35.
3. Add 7 to get 4x = 20, then divide by 4 to obtain x = 5.
4. Subtract 2 to get x/3 = 6, then multiply by 3 to obtain x = 18.
5. At x = 3, the left side is 16, not 17. Solving gives 5x = 16 and x = 16/5. Substitution gives 5(16/5) + 1 = 17.
6. Summary
- A solution makes the original equation true.
- Use inverse operations on both sides.
- Include the sign when identifying a coefficient.
- Exact fractions are valid solutions unless context restricts the answer.