Linear equations
Solve for x when x appears once or on both sides.
Pen and paper is fine · no calculator needed why?
Opens at level 16. You're level 1. You can read and practice here now.
the idea
An equation is a balance. Do the same thing to both sides and it stays true.
To solve, undo what was done to x, in reverse order. In 3x + 5 = 26, x was multiplied by 3 and then 5 was added. So take away the 5 first, then divide by 3.
Always finish the same way: put your answer back in and check that both sides match.
techniques
Undo in reverse order
- Undo the number added to or taken from the x term, on both sides.
- Divide both sides by the number in front of x.
- Check: put your answer back into the original equation.
worked example
Solve for x: 4x − 7 = 21
- Add 7 to both sides: 4x = 28.
- Divide both sides by 4: x = 7.
- Check: 4 × 7 − 7 = 21.
Answer: 7
Gather x on one side
- Multiply out any brackets. Multiply every term inside: 2(x − 3) is 2x − 6.
- Take one x term away from both sides. Taking the smaller one avoids minus signs.
- Move the plain numbers to the other side the same way.
- Divide by the number in front of x.
worked example
Solve for x: 3(x + 2) = x + 14
- Multiply out: 3x + 6 = x + 14.
- Take x from both sides: 2x + 6 = 14.
- Take 6 from both sides: 2x = 8.
- Divide by 2: x = 4.
Answer: 4
Fee plus rate
- Write it as fee + rate × hours = total.
- Take the fee off the total first.
- Divide what is left by the rate.
worked example
An electrician charges an $80 call-out fee plus $55 an hour. The bill is $355. How many hours did the job take?
- 80 + 55h = 355
- 55h = 355 − 80 = 275
- h = 275 ÷ 55 = 5
Answer: 5
watch out for
- Moving a number across without changing its sign: + 5 on one side becomes − 5 on the other.
- Multiplying only the first term in a bracket: 2(x − 3) is 2x − 6, not 2x − 3.
- Dividing the whole bill by the rate before taking off the fixed fee.
practice
Two-step equations
worked example
Solve for x: −2x − 10 = 10
Answer: -10
- Add 10 to both sides: −2x = 20.
- Divide by −2: x = −10.
x on both sides
worked example
Solve for x: −2x − 14 = 7x − 59
Answer: 5
- Take 7x from both sides: −9x − 14 = −59.
- Add 14 to both sides: −9x = −45.
- Divide by −9: x = 5.
Equations with brackets
worked example
Solve for x: 2(x + 5) = 4(x − 1)
Answer: 7
- Distribute: 2x + 10 = 4x − 4.
- Take 4x from both sides: −2x + 10 = −4.
- Take 10 from both sides: −2x = −14.
- Divide by −2: x = 7.
Fixed fee plus rate
worked example
A tool-hire shop charges a $44 delivery fee plus $40 per day for a floor sander. The hire cost $244 in total. For how many days was the sander hired?
Answer: 5
- Fee plus rate times days: 44 + 40d = 244.
- Take off the fee: 40d = 244 − 44 = 200.
- Divide by 40: d = 5.