What is an equation?
An equation is a statement that two expressions are equal. It always has an equals sign (=) and usually contains a variable: a letter, such as x or y, standing for an unknown number.
2x + 3 = 11
The expression on the left (2x + 3) is the LHS and the one on the right (11) is the RHS. The value of the variable that makes LHS = RHS is the solution. Here it is x = 4, because 2 × 4 + 3 = 11.
An expression like 2x + 3 on its own, with no equals sign, is not an equation.
An equation is a balance
Think of an equation as a weighing balance in equilibrium. It stays balanced if you do the same thing to both sides:
add the same number to both sides,
subtract the same number from both sides,
multiply both sides by the same number,
divide both sides by the same non-zero number.
Example. Solve x − 7 = 12.
Add 7 to both sides: x − 7 + 7 = 12 + 7, so x = 19.
Example. Solve 3y + 5 = 26.
Subtract 5 from both sides: 3y = 21.
Divide both sides by 3: y = 7.
Example. Solve m/4 = 6. Multiply both sides by 4: m = 24.
Transposing: a shortcut
Moving a term to the other side of the equals sign and changing its operation is called transposing. It is just a quick way of doing the same thing to both sides.
Operation on one side | becomes on the other side |
|---|---|
+ 5 | − 5 |
− 5 | + 5 |
× 3 | ÷ 3 |
÷ 3 | × 3 |
Example. Solve 4(p − 2) = 20.
Divide by 4 (transpose × 4): p − 2 = 5.
Add 2 (transpose − 2): p = 7.
Always check your answer
Substitute the solution back into the original equation. For 4(p − 2) = 20 with p = 7: LHS = 4 × (7 − 2) = 4 × 5 = 20 = RHS. ✓
From words to equations
Many problems are easier once they are written as an equation.
Words | Equation |
|---|---|
A number increased by 8 is 15 | x + 8 = 15 |
Three times a number is 27 | 3x = 27 |
Half of a number minus 4 is 6 | x/2 − 4 = 6 |
5 more than twice a number is 21 | 2x + 5 = 21 |
Example. Raju's father's age is 5 years more than three times Raju's age. His father is 44 years old. How old is Raju?
Let Raju's age be y years. Then 3y + 5 = 44, so 3y = 39 and y = 13. Raju is 13 years old.
Example. The sum of three consecutive whole numbers is 51. Find them.
Let the numbers be n, n + 1 and n + 2. Then 3n + 3 = 51, so 3n = 48 and n = 16. The numbers are 16, 17 and 18.
Tips
Decide what the variable stands for and write it down.
Undo operations in the reverse order: first undo the adding or subtracting, then the multiplying or dividing.
Check the answer, and make sure it makes sense (ages and lengths cannot be negative).