Simple Equations & Formulae

<p>An equation is like a balance scale — whatever is on one side must equal what is on the other. Solving an equation means finding the value of the unknown letter (often called x or n). Formulae work the same way but describe a general rule you can use again and again.</p>

<h2>Understanding Equations</h2>

<p>The equation 3x + 5 = 20 asks: "what number, when multiplied by 3 and then added to 5, gives 20?" Your job is to find x.</p>

<h2>Solving One-Step Equations</h2>

<p>Use the opposite (inverse) operation to undo what has been done to x.</p>

<p><strong>Example:</strong> x + 7 = 15. To undo the +7, subtract 7 from both sides: x = 15 − 7 = 8.</p>

<p><strong>Example:</strong> 4x = 28. To undo the ×4, divide both sides by 4: x = 28 ÷ 4 = 7.</p>

<h2>Solving Two-Step Equations</h2>

<p>Work in reverse order — undo addition or subtraction first, then multiplication or division.</p>

<p><strong>Worked Example:</strong> 3x + 5 = 20</p>

<p>Step 1: Subtract 5 from both sides → 3x = 15<br>

Step 2: Divide both sides by 3 → x = 5<br>

Check: 3(5) + 5 = 15 + 5 = 20. ✓</p>

<p><strong>Worked Example:</strong> (x − 4) / 2 = 6</p>

<p>Step 1: Multiply both sides by 2 → x − 4 = 12<br>

Step 2: Add 4 to both sides → x = 16<br>

Check: (16 − 4) / 2 = 12 / 2 = 6. ✓</p>

<h2>Equations with x on Both Sides</h2>

<p><strong>Worked Example:</strong> 5x + 2 = 3x + 10</p>

<p>Step 1: Subtract 3x from both sides → 2x + 2 = 10<br>

Step 2: Subtract 2 from both sides → 2x = 8<br>

Step 3: Divide both sides by 2 → x = 4<br>

Check: 5(4) + 2 = 22 and 3(4) + 10 = 22. ✓</p>

<h2>Using Formulae</h2>

<p>A formula lets you calculate one value when you know others. For example, the area of a rectangle is A = l × w. If l = 8 cm and w = 5 cm, then A = 8 × 5 = 40 cm².</p>

<p><strong>Worked Example:</strong> A taxi charges a fixed fee of £3 plus £2 per mile. The formula is C = 2m + 3, where C is the cost and m is the number of miles. What is the cost for 7 miles?</p>

<p>C = 2(7) + 3 = 14 + 3 = £17.</p>

<h2>Writing Your Own Equations</h2>

<p><strong>Worked Example:</strong> I think of a number, double it, and add 9. The answer is 25. What is my number?</p>

<p>Write the equation: 2n + 9 = 25. Solve: 2n = 16, n = 8. The number is 8.</p>

<h2>Function Machines</h2>

<p>A function machine shows operations applied in order. Input → ×3 → +2 → Output. If the input is 5, the output is 5 × 3 + 2 = 17. To work backwards (given output, find input), reverse the operations: 17 → −2 → ÷3 → 5.</p>

<h2>Practice Tips</h2>

<ul>

<li>Always do the same operation to BOTH sides of the equation — that keeps it balanced</li>

<li>Check your answer by substituting it back into the original equation</li>

<li>For word problems, start by choosing a letter and writing what it stands for</li>

<li>Remember: whatever was done last is undone first (like taking off layers of clothing)</li>

</ul>