What is a linear equation?
An equation is a statement that two expressions are equal. It is linear in one variable if it has only one variable and its highest power is 1.
2x + 5 = 13, 3(y − 1) = y + 7 and x/3 + 1 = 5 are linear equations.
x² + 1 = 10 is not linear (power 2), and x + y = 5 has two variables.
A linear equation in one variable has exactly one solution: the value of the variable that makes both sides equal.
The balance rule
An equation is like a balanced scale. You may add, subtract, multiply or divide both sides by the same number (not dividing by zero), and it stays balanced.
A shortcut called transposing follows from this: a term can be moved to the other side by changing its sign. A factor multiplying the whole side moves across as a divisor, and vice versa.
Example. 3x − 7 = 11
3x = 11 + 7 = 18, so x = 18 ÷ 3 = 6.
Check: 3 × 6 − 7 = 11. ✓ Always check by substituting back.
Variable on both sides
Collect the variable terms on one side and the numbers on the other.
Example. 5x + 4 = 2x + 19
5x − 2x = 19 − 4, so 3x = 15 and x = 5.
Brackets
Expand first, then solve.
Example. 4(x − 3) = 2(x + 5)
4x − 12 = 2x + 10, so 2x = 22 and x = 11.
Fractions
Multiply every term on both sides by the LCM of the denominators to clear the fractions.
Example. x/2 + x/3 = 10
The LCM of 2 and 3 is 6: 3x + 2x = 60, so 5x = 60 and x = 12.
Example. (2x − 1)/3 − (x − 2)/4 = 1
Multiply by 12: 4(2x − 1) − 3(x − 2) = 12, so 8x − 4 − 3x + 6 = 12, which gives 5x = 10 and x = 2.
Note the bracket around (x − 2): forgetting it changes −3 × (−2) = +6 into −6.
Equations reducible to linear form
Some equations have the variable in a denominator. Cross-multiply to make them linear.
Example. (x + 2)/(x − 1) = 4/3
3(x + 2) = 4(x − 1), so 3x + 6 = 4x − 4 and x = 10.
Word problems
Read carefully and decide what the unknown is. Call it x.
Write the other quantities in terms of x.
Form an equation from the condition given.
Solve, and check the answer makes sense in the problem.
Example: consecutive numbers. The sum of three consecutive integers is 72. Let them be x, x + 1 and x + 2. Then 3x + 3 = 72, so x = 23. The numbers are 23, 24 and 25.
Example: ages. A mother is 3 times as old as her son. In 12 years she will be twice as old as he will be. Let the son be x years old; the mother is 3x. In 12 years: 3x + 12 = 2(x + 12), so 3x + 12 = 2x + 24 and x = 12. The son is 12 and the mother 36.
Example: digits. The digits of a two-digit number add up to 9. When the digits are reversed, the number increases by 27. Let the units digit be x; the tens digit is 9 − x. The number is 10(9 − x) + x = 90 − 9x, and the reversed number is 10x + (9 − x) = 9x + 9. So 9x + 9 = 90 − 9x + 27, which gives 18x = 108 and x = 6. The number is 36 (63 − 36 = 27 ✓).
Summary
Do the same thing to both sides; transposing changes the sign of a term.
Clear brackets by expanding and fractions by multiplying by the LCM.
Cross-multiply when the variable is in a denominator.
In word problems, define x clearly and check the answer.