Linear Equations

A linear equation is an algebraic equation where each term is either a constant or a single variable raised to the first power.

What is a Linear Equation?

A linear equation in one variable has the form:

$ax + b = 0$

where $a \neq 0$ . Solving for $x$ gives:

$x = -\frac{b}{a}$

Standard Form

ax + b = 0 \implies x = -\dfrac{b}{a}

where a ≠ 0. The solution is unique.

Solving Linear Equations

Follow these steps to solve a linear equation:

1
Simplify both sides
Expand brackets and collect like terms on each side of the equation.
2
Move variables to one side
Add or subtract terms so all variable terms are on the left and constants on the right.
3
Isolate the variable
Divide both sides by the coefficient of the variable.
4
Check your solution
Substitute your answer back into the original equation to verify it is correct.

Solving 3x + 7 = 19

Given: $3x + 7 = 19$

Step 1: Subtract 7 from both sides: $3x = 12$

Step 2: Divide both sides by 3: $x = 4$

Check: $3(4) + 7 = 12 + 7 = 19$ ✓

Linear Equations in Two Variables

A linear equation in two variables $x$ and $y$ has the form:

$ax + by = c$

This represents a straight line in the coordinate plane. The slope-intercept form is:

$y = mx + b$

where $m$ is the slope and $b$ is the y-intercept.

Slope Formula

m = \dfrac{y_2 - y_1}{x_2 - x_1}

The slope m measures the rate of change: rise over run between two points.

Common Forms

Form	Equation	When to Use
Standard	$ax + by = c$	General form
Slope-intercept	$y = mx + b$	Known slope and y-intercept
Point-slope	$y - y_1 = m(x - x_1)$	Known point and slope

Common Mistakes

Forgetting to apply operations to both sides: Whatever you do to one side, do to the other.
Sign errors when moving terms: Moving $+7$ to the right gives $-7$ .
Dividing by zero: Check that $a \neq 0$ before dividing.

Tip

Always verify your solution by substituting back into the original equation.

Real-World Applications

Real world: Taxi fare

A taxi charges a €2.50 base fare plus €1.20 per km. A journey costs €10.30 total. How far was the journey? Let $x$ = distance in km: $1.20x + 2.50 = 10.30 \implies 1.20x = 7.80 \implies x = 6.5$ km.

Real world: Mobile phone bill

A mobile plan costs €8 per month plus €0.05 per minute of calls. Last month's bill was €18. How many minutes were used? Let $m$ = minutes: $0.05m + 8 = 18 \implies 0.05m = 10 \implies m = 200$ minutes.

Real world: Recipe scaling

A recipe for 4 people uses $x$ grams of rice. Scaling it up for 7 people requires 630 g in total. The equation $\frac{7}{4}x = 630$ gives $x = 360$ g — the amount for the original 4-person version.

Tip

Where do you see this? Taxi and ride-share receipts, electricity and phone bills, cooking and recipe scaling apps, currency exchange calculations.

Practice Problems

Solve: 5x − 3 = 17

Show answer

5x = 20, so x = 4

Solve: 2(x + 4) = 14

Show answer

2x + 8 = 14 → 2x = 6 → x = 3

Find the slope of the line through (1, 2) and (3, 8)

Show answer

m = (8 − 2)/(3 − 1) = 6/2 = 3