How do apps work out taxi fares or how do shops calculate discounts? These everyday problems are often solved using linear equations. Think of it like being a detective trying to find a hidden value. This guide will help you crack the code!
What’s a Linear Equation?
An equation is a statement that two things are equal, like a perfectly balanced set of scales. A linear equation is a specific type where the highest power of the unknown variable (like $x$) is 1. When graphed, it always forms a straight line!
The goal is to find the value of the unknown that makes the equation true.
How to Solve Linear Equations
The golden rule is: whatever you do to one side, you must do to the other. We use inverse operations (like addition and subtraction, or multiplication and division) to “undo” the equation and find the unknown.
Type 1: Simple Equations
To solve $x + 5 = 12$, we want to get $x$ by itself. The inverse of adding 5 is subtracting 5.
$x + 5 – 5 = 12 – 5$
$x = 7$
To solve $3x = 15$, we undo multiplying by 3 by dividing by 3.
$\frac{3x}{3} = \frac{15}{3}$
$x = 5$
Type 2: Unknowns on Both Sides
To solve $6x – 5 = 2x + 15$, first gather the $x$ terms on one side. It’s often easiest to move the smaller one.
- Subtract $2x$ from both sides: $4x – 5 = 15$
- Add 5 to both sides: $4x = 20$
- Divide by 4: $x = 5$
Type 3: Equations with Brackets
To solve $3(x + 4) = 21$, your first step is always to expand the bracket.
- Expand: $3x + 12 = 21$
- Subtract 12: $3x = 9$
- Divide by 3: $x = 3$
Type 4: Combined Equations
To solve $2(x – 3) = 4x + 2$, combine all the steps.
- Expand: $2x – 6 = 4x + 2$
- Subtract $2x$: $-6 = 2x + 2$
- Subtract 2: $-8 = 2x$
- Divide by 2: $x = -4$
Finding Solutions with a Graph
You can also find an approximate solution by treating each side of the equation as a straight line. The solution is the x-coordinate of the point where the two lines intersect (cross).
Example: Solve $2x + 3 = 9$ graphically
We plot two lines: $y = 2x + 3$ and the horizontal line $y = 9$. The graph shows they intersect at the point $(3, 9)$. The x-coordinate of this point is 3, so the solution is $x = 3$.
Tutor Insights
🤔 Common Misunderstandings
- Sign Errors: Forgetting to change the sign when moving a term across the equals sign (e.g., moving $+5$ across means it becomes $-5$).
- Incorrectly Expanding Brackets: Writing $2(x+3)$ as $2x+3$ instead of $2x+6$. Remember to multiply everything inside!
- Not Doing the Same to Both Sides: This unbalances the equation and leads to the wrong answer.
📝 Common Exam Mistakes
- Simple Arithmetic Errors: Even with a perfect method, a small calculation mistake can cost marks.
- Not Showing Working: You can get method marks for logical steps even if your final answer is wrong.
- Forgetting to Check: If you have time, substitute your answer back into the original equation to see if it works.
Practice Questions
- Solve: $x + 12 = 20$
- Solve: $6y = 42$
- Solve: $4a – 5 = 19$
- Solve: $2(m + 7) = 22$
- Solve: $5x – 3 = 2x + 9$
- I think of a number, multiply it by 4, then subtract 7. The answer is 13. Form an equation and solve it.
Show Answers
- $x = 8$
- $y = 7$
- $a = 6$
- $m = 4$
- $x = 4$
- Equation: $4n – 7 = 13$. Solution: $4n = 20$, so $n = 5$.
FAQs
Q: Why is it called a “linear” equation?
A: Because when you plot the relationship on a graph, it always forms a perfect straight line. The word “line” is right there in “linear”!
Q: Can the answer to an equation be negative?
A: Absolutely! The unknown variable $x$ can represent any number, including negative numbers, zero, or fractions. Don’t be put off if your answer isn’t a positive whole number.
Want to Master Solving Equations?
A qualified GCSE maths tutor can walk you through every type of linear equation — and build the algebra skills you need for the whole exam.
Find a GCSE Maths Tutor
