courses › Algebra

Linear equations

level 16 course

Solve for x when x appears once or on both sides.

Pen and paper is fine · no calculator needed why?

Opens at level 16.

Builds on: Expressions & one-step equations (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

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

x appears once.

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

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

x appears on both sides, or inside brackets.

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

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

A fixed fee plus a rate per hour, per visit or per item.

Write it as fee + rate × hours = total.
Take the fee off the total first.
Divide what is left by the rate.

worked example

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

Tips by skill

TipTwo-step equations: Undo the number added or taken away first. Then divide by the number in front of x, sign included.
Tipx on both sides: Get every x on one side first: take one of the x terms away from both sides.
TipEquations with brackets: Multiply out the brackets first, and multiply every term inside them.
TipFixed fee plus rate: Take the fixed fee off the total before you divide by the rate.

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.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

Two-step equations par 10 s not tried yet
worked example
Solve for x: −2x − 10 = 10

Answer: -10
1. Add 10 to both sides: −2x = 20.
2. Divide by −2: x = −10.
x on both sides par 16 s not tried yet
worked example
Solve for x: −2x − 14 = 7x − 59

Answer: 5
1. Take 7x from both sides: −9x − 14 = −59.
2. Add 14 to both sides: −9x = −45.
3. Divide by −9: x = 5.
Equations with brackets par 19 s not tried yet
worked example
Solve for x: 2(x + 5) = 4(x − 1)

Answer: 7
1. Distribute: 2x + 10 = 4x − 4.
2. Take 4x from both sides: −2x + 10 = −4.
3. Take 10 from both sides: −2x = −14.
4. Divide by −2: x = 7.
Fixed fee plus rate par 30 s not tried yet
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
1. Fee plus rate times days: 44 + 40d = 244.
2. Take off the fee: 40d = 244 − 44 = 200.
3. Divide by 40: d = 5.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

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