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.
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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.
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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.
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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.