Get the letter on its own
To solve an equation is to find the number the letter stands for. In x + 2 = 7, x is not on its own: 2 has been added to it. The aim is to change the equation, one step at a time, until x stands alone on one side and a number stands on the other.
An equation is like a level balance. Take the same amount off both pans and they stay level. Take it off one pan only and the scales tip, because the two sides are no longer equal. So whatever you do to one side of an equation, you must do to the other.
Take 2 off both pans. The left keeps x, and the right keeps 7 − 2 = 5.
Undo the addition
In x + 2 = 7, 2 is added to x. Subtraction undoes addition, so subtract 2 from both sides: x + 2 − 2 = 7 − 2, which leaves x = 5.
Check the answer by putting it back into the equation you started with: 5 + 2 = 7. Both sides agree, so x = 5 is right.
x is alone and the pans are still level, so x = 5.
Two steps
Most equations need more than one step. In 3x + 2 = 14, x has been multiplied by 3, and then 2 has been added. Undo them in the reverse order: first the addition, then the multiplication.
Subtract 2 from both sides: 3x + 2 − 2 = 14 − 2, so 3x = 12.
Both pans lose 2, which leaves 3x on the left and 12 on the right.
Then undo the multiplication
3x means 3 × x, three lots of x, and together they make 12. Division undoes multiplication, so divide both sides by 3: , which gives x = 4.
Check it in the equation you started with: 3 × 4 + 2 = 12 + 2 = 14.
3x is three equal lots of x, and together they make 12. Sharing 12 into 3 equal lots puts 4 in each, so x = 4.
Both pans, or one
The balance below holds 3x + 7 = 22: three x blocks and 7 units on the left, and 22 units on the right. Take units off both pans and the beam stays level at every step. Take them off the left pan only and it tips.
3x + 7 = 22 balances: the three x blocks and 7 units weigh what 22 units weigh; take the 7 off both pans and the x blocks weigh 15 alone
Take 7 off both pans, then read x
Take the 7 units off both pans. The three x blocks then weigh 15 on their own, so 3x = 15 and . Check: 3 × 5 + 7 = 22.
The usual mistakes
Stopping one step early. In 3x + 2 = 14, the answer is not 12. 12 is what 3x equals, three lots of x together, so divide by 3 to find one x.
Undoing the 3 the wrong way. 3x means 3 times x, so the 3 is undone by dividing by 3. Subtracting 3 or adding 3 does not undo a multiplication.
Changing one side only. Taking 2 off the left side and not the right tips the balance: the new equation is not true, and neither is its answer.
Worked example: A Taxi Fare with a Fixed Charge and a Charge per Kilometer
Question A taxi company charges a fixed $5 for every trip and then $3 for each kilometer traveled. (a) Find the fare for a trip of 12 km. (b) The fare for another trip is $62. How long is that trip?
1.Let the length of a trip be d km. The fare is the fixed charge plus $3 for each kilometer, which is 5 + 3d dollars.
A trip of d km costs the fixed $5 and $3 for each kilometer: 5 + 3d dollars. 2.(a) For d = 12 the fare is 5 + 3 × 12 = 5 + 36 = $41.
(a) For 12 km the fare is 5 + 3 × 12 = $41. 3.For the other trip the fare is $62, so 5 + 3d = 62.
A fare of $62 gives the equation 5 + 3d = 62. 4.Subtract 5 from both sides: 3d = 57. Divide both sides by 3: d = 19.
Subtract 5 from both sides, then divide both sides by 3: d = 19. 5.(b) The trip is 19 km long. Check: 5 + 3 × 19 = 5 + 57 = 62.
(b) The trip is 19 km long.
Answer: (a) $41; (b) 19 km
Common mistakes
- Dividing the whole fare by 3, 62 ÷ 3. The fixed charge of $5 is not paid per kilometer, so it is subtracted first and only the remaining $57 is divided by 3.
- Working out (5 + 3) × 12 = 96 in part (a). The fixed charge is paid once, not once for every kilometer, so the fare is 5 + 3 × 12.
More negative numbers and linear equations problems, worked step by step →