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§ Expressions & Algebra

Introduction to Equations

§ Expressions & Algebra

Introduction to Equations

CCSS.6.EECCSS.7.EE3 min read

An equation is a mathematical statement that shows two expressions are equal, separated by an equals sign. The goal when solving equations is to find the value of the unknown variable (usually represented by x) that makes the statement true. In Year 7 mathematics, students begin with simple linear equations in one variable, which form the foundation for all algebraic problem-solving.

§ 01

Why it matters

Equations appear throughout real-world problem-solving and form the backbone of mathematical modelling. A plumber might use the equation 45x + 25 = 115 to work out how many hours (x) a job took when the total bill was £115. Shop managers use equations like 3x = 24 to calculate how many £3 items they sold for £24 revenue. Scientists rely on equations to model everything from population growth to chemical reactions. In GCSE mathematics, equation-solving skills are essential for topics including simultaneous equations, quadratics, and coordinate geometry. The ability to manipulate equations systematically develops logical thinking patterns that extend beyond mathematics into computer programming, engineering, and financial planning. Students who master simple one-step equations in Year 7 build confidence for more complex algebraic manipulation required at GCSE level and beyond.

§ 02

How to solve introduction to equations

One-Step Equations

  • An equation has an unknown (x) and an equals sign.
  • Use the inverse operation to isolate x.
  • Addition ↔ subtraction; multiplication ↔ division.
  • Check by substituting your answer back.

Example: x + 7 = 12 → x = 12 − 7 = 5.

§ 03

Worked examples

Beginner§ 01

x + 9 = 17. What is x?

Answer: 8

  1. Subtract 9 from both sides → x = 17 − 9 — To isolate x, subtract the number being added.
  2. Calculate → x = 8 — 17 − 9 = 8.
Easy§ 02

x − 2 = 7. What is x?

Answer: 9

  1. Add 2 to both sides → x = 7 + 2 — To undo subtraction, add the same number to both sides.
  2. Calculate → x = 9 — 7 + 2 = 9.
Medium§ 03

9x = 72. What is x?

Answer: 8

  1. Divide both sides by 9 → x = 72 ÷ 9 — To isolate x, divide by the coefficient 9.
  2. Calculate → x = 8 — 72 ÷ 9 = 8.
§ 04

Common mistakes

  • Adding instead of subtracting when solving x + 5 = 12, giving x = 17 instead of x = 7
  • Forgetting to apply the same operation to both sides, writing 3x = 15 as x = 15 instead of x = 5
  • Mixing up the direction of subtraction in problems like x - 4 = 9, calculating x = 4 - 9 = -5 instead of x = 9 + 4 = 13
§ 05

Frequently asked questions

What is the difference between an expression and an equation?
An expression like 2x + 5 has no equals sign and represents a value. An equation like 2x + 5 = 13 has an equals sign and states that two expressions are equal. Expressions are simplified, whilst equations are solved to find the unknown variable.
How do you check if your answer to an equation is correct?
Substitute the answer back into the original equation. For x + 3 = 8 with answer x = 5, check: 5 + 3 = 8. Since 8 = 8 is true, the answer is correct. If the left and right sides don't match, the solution is wrong.
Why do you do the same thing to both sides of an equation?
An equation is like a balance scale — both sides are equal. If you add 3 to one side but not the other, the balance breaks. Performing the same operation on both sides maintains the equality whilst isolating the unknown variable.
What does 'inverse operation' mean in equation solving?
The inverse operation undoes what was done to the variable. Addition's inverse is subtraction, multiplication's inverse is division. If x is increased by 6, subtract 6 to reverse it. If x is multiplied by 4, divide by 4 to undo it.
When do you use division instead of multiplication to solve an equation?
Use division when the unknown is multiplied by a number. In 7x = 35, x is multiplied by 7, so divide both sides by 7 to get x = 5. Division isolates x by removing the coefficient attached to it.
§ 06

See also

EquationVariableExpressionMeanCoefficient
§ 06

Where to next?

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