Simple equations

An equation is two expressions joined by an equal sign. Solving an equation means finding the value of the variable that makes both sides equal.

Idea

Compare the two:

• $x + 4 = 9$ , an equation; you must find $x$.
• $x + 4$ , just an expression; no unknown to solve.

To solve $x + 4 = 9$, ask: what number, when increased by $4$, gives $9$? Clearly $x = 5$. We say the equation is true at $x = 5$ and false at every other value.

A cleaner method: apply the balance rule. Whatever you do to one side, do to the other side, and the equation stays balanced.

• To remove $+4$ from the left, subtract $4$ from both sides: $x + 4 - 4 = 9 - 4 \implies x = 5.$
• To remove a multiplier, divide both sides by the same number. For $3x = 15$: $\frac{3x}{3} = \frac{15}{3} \implies x = 5.$

Use these rules step by step. The aim is to isolate the variable on one side.

Check your solution by substituting back into the original equation. For $x + 4 = 9$ with $x = 5$: $5 + 4 = 9$. ✓

Equations help us answer "find the unknown" problems from real life , Aman's age, the cost of a missing item, the side of a square given its perimeter, and so on.

Worked examples

Example 1. Solve $x + 7 = 12$.

Subtract $7$ from both sides: $x = 12 - 7 = 5$. Check: $5 + 7 = 12$. ✓

Example 2. Solve $4y = 28$.

Divide both sides by $4$: $y = 28 \div 4 = 7$. Check: $4(7) = 28$. ✓

Example 3. Solve $2x + 5 = 17$.

Subtract $5$: $2x = 12$. Divide by $2$: $x = 6$. Check: $2(6) + 5 = 12 + 5 = 17$. ✓

Example 4. A book and a pen together cost $\rupee 80$. The book costs $\rupee 50$. Find the pen's cost using an equation.

Let pen cost $\rupee p$. Then $p + 50 = 80$. So $p = 30$. The pen costs $\rupee 30$.

Try it yourself

1. Solve $x - 3 = 10$.
2. Solve $5y = 35$.
3. Solve $a + 12 = 20$.
4. Solve $3n - 4 = 11$.
5. Solve $\dfrac{x}{4} = 6$.
6. Solve $2(x + 3) = 14$.
7. The perimeter of a square is $36$ cm. Form an equation and find the side.
8. The sum of a number and $7$ is $21$. Form and solve an equation.

Activity

Make a "balance scale" using two equal-sized paper plates and a ruler resting on a pencil. Place equal coins on both sides , that's the equation. Now solve a real equation by adding/removing coins equally from both sides until only the unknown remains.