Forming and solving equations is like being a detective: you take clues from a real-world problem, translate them into maths, crack the code to find the unknown, and then interpret your result. It’s a crucial skill for your GCSE and for solving problems in everyday life.
The 3-Step Detective Method
1. Form the Equation
Translate the words or diagram into a mathematical statement. Identify the unknown value and represent it with a variable (like $x$).
2. Solve the Equation
Use your algebra skills, like inverse operations, to find the value of the unknown variable. Remember to keep the equation balanced!
3. Interpret the Solution
Don’t just stop at $x=5$! Explain what your answer means in the context of the original problem, including any units.
Worked Examples
Example 1: Word Problem
Sarah thinks of a number. She multiplies it by 4 and adds 7. Her answer is 31. What was the number?
- Form the equation: Let the number be $x$. The equation is $4x + 7 = 31$.
- Solve:
$4x = 31 – 7$
$4x = 24$
$x = 24 \div 4 = 6$. - Interpret: The number Sarah thought of was 6.
Example 2: Perimeter Problem
A rectangle has a perimeter of 30 cm. Its length is $(x+5)$ and its width is $x$. Find the value of $x$ and the length of each side.
- Form the equation: Perimeter = $x + (x+5) + x + (x+5) = 4x + 10$. The equation is $4x + 10 = 30$.
- Solve:
$4x = 30 – 10$
$4x = 20$
$x = 20 \div 4 = 5$. - Interpret: $x=5$. The width is 5 cm and the length is $5+5 = \mathbf{10 \text{ cm}}$.
Tutor Insights
🤔 Common Misunderstandings
- Mixing up expressions and equations: Equations have equals signs; expressions don’t. When forming an equation, you need to set your expression equal to the result given in the question.
- Forgetting to interpret the solution: Solving for $x$ is only half the job! You need to state what that value means in the context of the problem.
📝 Common Exam Mistakes
- Sign errors when moving terms across the equals sign.
- Not doing the same to both sides. If you add 2 on the left, you must add 2 on the right to keep it balanced.
- Not reading the question fully: A question might ask for “the value of $x$” AND “the area”. Students often stop after finding $x$.
- Not showing your working. You must show how you formed and solved the equation to get full marks.
Practice Questions
- A number is multiplied by 5, and then 3 is subtracted. The answer is 37. What is the number?
- A rectangular garden has a width of $w$ metres and a length of $w+6$ metres. The perimeter is 40 metres. Find the length and width of the garden.
- A triangle has angles measuring $x+20$, $3x$, and $2x-10$. Find the size of each angle.
Show Answers
- Equation: $5x-3=37$. Solution: $5x=40 \implies x=8$. The number is 8.
- Equation: $4w+12=40$. Solution: $4w=28 \implies w=7$. The width is 7m and the length is 13m.
- Equation: $(x+20)+3x+(2x-10)=180 \implies 6x+10=180$. Solution: $6x=170 \implies x = 28\frac{1}{3}$. The angles are $48\frac{1}{3}^\circ$, $85^\circ$, and $46\frac{2}{3}^\circ$.
FAQs
How do I know what letter to use for the variable?
You can use any letter you like! ‘x’ is common, but it’s often helpful to use a letter that relates to the problem, like ‘w’ for width or ‘a’ for age. This can make your equations easier to understand.
What if I get a negative or fractional answer for $x$?
Mathematically, that’s perfectly fine! However, you must always interpret it in the context of the original problem. If $x$ represents a length, it can’t be negative. If $x$ represents a number of people, it can’t be a fraction. Always check if your answer makes sense for the situation.
