Reference · module 2
Linear equations
Everything this module explains, on one page.
Equations as a balance
Solve one-step equations by doing the same to both sides.
- An equation says two sides are equal, like a balance. Whatever you do to one side you must do to the other, or it stops being true.
- To free x, undo what is done to it with the opposite operation. In x + 7 = 12, subtract 7 from both sides: x = 5. In 3x = 18, divide both by 3: x = 6.
- Always check by putting the answer back: 3 × 6 is 18, so x = 6 is right. A check takes five seconds and catches most mistakes.
The mistake you are about to make
x + 7 = 12, x = 19
x + 7 = 12, x = 5
The 7 and 12 are both on screen, so adding them feels like progress. To remove +7 you subtract 7 from both sides, which gives 5.
Two-step equations
Solve equations where x is both multiplied and added to.
- In 2x + 5 = 17 two things happen to x: it is doubled, then 5 is added. Undo them in reverse order — like taking off shoes before socks.
- First remove the added number: 2x + 5 = 17 becomes 2x = 12. Then divide: x = 6. Check: 2 × 6 + 5 is 17.
- Solutions can be negative or fractions. In 4x + 10 = 2, subtract 10 to get 4x = −8, so x = −2. A negative answer is not a sign of a mistake.
The mistake you are about to make
2x + 5 = 17, x = 11
2x + 5 = 17, x = 6
Dividing 17 by 2 or removing 5 alone feels like enough. Both steps are needed, and in order: minus 5 first, then divide by 2.
x on both sides
Solve equations with brackets or x on both sides.
- When x appears on both sides, move all the x terms to one side first. In 5x + 2 = 3x + 10, subtract 3x from both: 2x + 2 = 10, so x = 4.
- With brackets, expand them first and then solve as usual: 3(x − 2) = 12 becomes 3x − 6 = 12, so 3x = 18 and x = 6.
- A word problem becomes an equation once you name the unknown. A number doubled plus 7 is 25: 2n + 7 = 25, so n = 9.
The mistake you are about to make
5x + 2 = 3x + 10, 8x = 12
5x + 2 = 3x + 10, 2x = 8
Terms on opposite sides look like they should be added together. Moving 3x across the equals sign changes its sign, so it is subtracted.