Two-step equations, inverse operations, balance method
Middle School Bridge • 6-8
An equation is like a balanced scale: whatever sits on one side must equal what sits on the other. When one of the values is unknown, your job is to figure out what number makes the equation true. This lesson teaches you to solve equations using inverse operations -- the fundamental skill of algebra.
An equation is a mathematical statement that two expressions are equal, such as 2x + 3 = 11. The value of x that makes the equation true is called the solution.
Think of the equals sign as the center of a balance scale. To keep it balanced, any operation you perform on one side, you must also perform on the other side. You use inverse operations (opposites) to isolate the variable:
| Operation | Inverse |
|---|---|
| Addition (+) | Subtraction (-) |
| Subtraction (-) | Addition (+) |
| Multiplication (×) | Division (÷) |
| Division (÷) | Multiplication (×) |
Solve: x + 8 = 15.
Check: 7 + 8 = 15. Correct.
Solve: 3x - 5 = 16.
Check: 3(7) - 5 = 21 - 5 = 16. Correct.
Solve: x⁄4 + 2 = 9.
Check: 28 ÷ 4 + 2 = 7 + 2 = 9. Correct.
For multi-step equations, undo operations in reverse order. Since the expression was built by first multiplying and then adding/subtracting, undo the addition/subtraction first, then undo the multiplication/division. Think of it as peeling off layers from the outside in.
Forgetting to do the same thing to both sides. If you subtract 5 from the left side, you must also subtract 5 from the right side. Otherwise the equation becomes unbalanced and leads to a wrong answer.
1. Solve: x - 9 = 14.
Add 9 to both sides: x = 14 + 9 = 23. Check: 23 - 9 = 14.
2. Solve: 5x = 45.
Divide both sides by 5: x = 9. Check: 5(9) = 45.
3. Solve: 2x + 7 = 19.
Subtract 7: 2x = 12. Divide by 2: x = 6. Check: 2(6) + 7 = 19.
4. Solve: n⁄3 - 4 = 2.
Add 4: n⁄3 = 6. Multiply by 3: n = 18. Check: 18 ÷ 3 - 4 = 6 - 4 = 2.
5. Solve: 4x + 3 = -9.
Subtract 3: 4x = -12. Divide by 4: x = -3. Check: 4(-3) + 3 = -12 + 3 = -9.
To solve an equation, use inverse operations to isolate the variable, always performing the same operation on both sides. For two-step equations, undo addition or subtraction first, then undo multiplication or division. Always check your solution by substituting it back into the original equation.