Choose a variable side
When a variable appears on both sides, use addition or subtraction to collect all variable terms on one side. Choosing the side with the larger variable coefficient often keeps the final coefficient positive, but either side works if every operation is applied to both sides.
After the variable terms are together, collect the constants on the other side. Then divide by the remaining coefficient and substitute the result into the original equation to check it.
Notice what happens when variables cancel
Sometimes the variable terms cancel completely. If the remaining statement is always true, such as 5 = 5, every value is a solution. If it is false, such as 5 = 9, no value can make the original equation true.
Those outcomes are information, not mistakes. They describe whether the two sides represent the same expression for every input or can never represent the same value.
Remember these ideas
Key ideas
- Apply the same operation to both sides to preserve equality.
- Collect variable terms on one side and constants on the other.
- If variables cancel, inspect the remaining statement before deciding the solution set.
- Check a single solution in the original equation.
Make the reasoning visible
Worked example · 6x + 4 = 2x + 20
- 1
Subtract 2x from both sides: 4x + 4 = 20.
- 2
Subtract 4 from both sides: 4x = 16.
- 3
Divide both sides by 4: x = 4.
- 4
Check: 6(4) + 4 = 28 and 2(4) + 20 = 28, so the solution works.
Try the next step
Collect the variable terms
Solve 7y − 5 = 3y + 19, then check your result.
Why might moving the smaller variable term make the arithmetic easier even though another valid route exists?