Solving Linear Equations with One Variable
The goal when solving linear equations is to find the value of the unknown variable.
One-step equations
The simplest linear equations require one operation to solve. To solve x + 5 = 12: subtract 5 from both sides → x = 7. To solve 3x = 15: divide both sides by 3 → x = 5. The principle is always the same: perform the same operation on both sides to keep the equation balanced.
Multi-step equations
More complex linear equations require multiple steps. To solve 2x + 3 = 11: first subtract 3 from both sides (2x = 8), then divide by 2 (x = 4). When variables appear on both sides, collect them on one side first. For 5x − 2 = 3x + 6: subtract 3x from both sides → 2x − 2 = 6 → x = 4.
Equations with brackets
Expand brackets first, then solve. 3(x + 2) = 18 → 3x + 6 = 18 → 3x = 12 → x = 4. Checking the answer by substituting back into the original equation is always good practice. Algebra provides the full toolkit for working with equations.
Linear Equations with Two Variables
Linear equations in two variables (x and y) describe relationships between quantities and can be plotted as straight lines.
The form y = mx + c
Every straight line can be written as y = mx + c. Here m is the gradient (steepness) and c is the y-intercept — where the line crosses the y-axis. For y = 2x + 1: the gradient is 2 (rises 2 for every 1 across) and the y-intercept is 1.
Plotting linear equations
To plot a linear equation, choose two or three values of x, calculate the corresponding y values, plot the coordinate pairs, and join them with a straight line. For y = 3x − 2: substituting x = 0, 1, 2 gives y = −2, 1, 4. Connect the points.
Simultaneous linear equations
Two linear equations can be solved simultaneously to find values of x and y that satisfy both. Graphically, the solution is the point where two lines intersect. Algebraically, substitute one equation into the other or add/subtract them to eliminate a variable. Coordinates are essential for plotting linear equations on a graph.
Linear Equations in the Real World
Linear equations model constant-rate relationships found throughout science and everyday life.
Distance, speed, and time
The relationship d = vt (distance equals speed times time) is a linear equation. If you travel at 60 km/h, the distance after t hours is 60t — a linear relationship. Plotting distance against time gives a straight line whose gradient is the speed.
Money and budgeting
Linear equations model financial situations. If a phone costs £30 per month on a plan and has a £50 setup fee, the total cost after m months is 30m + 50. Solving linear equations with this model answers questions like 'after how many months does Plan A become cheaper than Plan B?'
Science and medicine
Drug dosage often scales linearly with body weight. Temperature conversion (Celsius to Fahrenheit: F = 9/5 × C + 32) is a linear equation. Physics equations for force, momentum, and Ohm's law (V = IR) are all linear relationships between variables.
Frequently asked questions
- What makes an equation 'linear'?
- An equation is linear when the variable appears only to the first power — no x², no √x, no 1/x. The graph of a linear equation in two variables is always a straight line. Equations like y = x² (a curve) or y = 1/x (a hyperbola) are not linear.
- What is the difference between an equation and an expression?
- An expression is a mathematical phrase with no equals sign — like 3x + 2. An equation states that two expressions are equal — like 3x + 2 = 11. You can simplify expressions, but you solve equations by finding the value of the variable that makes both sides equal.
- How do you solve linear equations with fractions?
- Multiply every term by the least common multiple (LCM) of the denominators to clear the fractions. For x/3 + 1 = 5: multiply all terms by 3 → x + 3 = 15 → x = 12. Clearing fractions converts the equation into a standard linear form without decimals.
- What happens when a linear equation has no solution or infinite solutions?
- If solving leads to a false statement (like 3 = 5), the equation has no solution — the lines are parallel and never intersect. If solving leads to a true statement (like 0 = 0), there are infinite solutions — the equations describe the same line. These cases arise when solving simultaneous linear equations.