Simple Equations Class 7: Forming, Solving & Word Problems

Here's a riddle: "I'm thinking of a number. If I add $7$ to it, I get $15$. What's my number?" You'd quickly say $8$. But what you just did, without realising it, is solve the equation $x + 7 = 15$.

Simple equations are one of the most exciting topics in Class 7 because they're your first real taste of algebra, the art of finding unknowns. In NCERT Class 7 Math (Chapter 4: Simple Equations), you'll learn to solve problems systematically using equations instead of guesswork. This skill is the foundation for everything you'll do in algebra for years to come!

An equation is a mathematical statement that says two things are equal, connected by the $=$ sign.

Examples of equations:
- $x + 5 = 12$
- $3y - 7 = 14$
- $2p + 1 = p + 8$

Not equations (these are expressions):
- $x + 5$ (no equals sign)
- $3y - 7$ (no equals sign)

The letter ($x$, $y$, $p$) is called the variable, and finding its value is called solving the equation. The value that makes the equation true is called the solution or root of the equation.

For example, in $x + 5 = 12$, the solution is $x = 7$ because $7 + 5 = 12$. If you tried $x = 6$, you'd get $6 + 5 = 11 \neq 12$, so $6$ is NOT a solution.

Before you can solve an equation, you need to set it up. Translating words into algebra is a crucial skill.

Here's a handy translation guide:

Example 1: "$6$ added to thrice a number gives $27$."

Example 2: "One-fifth of a number is $3$ more than $4$."

Example 3: "Ravi's age is $3$ years more than twice Sita's age. Ravi is $19$ years old."
Let Sita's age $= x$.

Practice forming equations from word statements until it feels natural. This is the hardest part for most students!

Think of an equation as a weighing balance. Both sides are perfectly balanced. To keep the balance, anything you do to one side, you must do to the other.

Example 1: Solve $x + 9 = 15$.

Subtract $9$ from both sides:

Example 2: Solve $3x = 21$.

Divide both sides by $3$:

Example 3: Solve $5x - 3 = 17$.

Step 1: Add $3$ to both sides.

Step 2: Divide both sides by $5$.

Verify: $5(4) - 3 = 20 - 3 = 17$. Correct!

The balancing method is intuitive because you can literally picture a scale staying balanced. It's the safest method when you're starting out.

Transposing is a shortcut for the balancing method. When you move a term from one side of the equation to the other, you change its sign:
- $+$ becomes $-$
- $-$ becomes $+$
- $\times$ becomes $\div$
- $\div$ becomes $\times$

Example 1: Solve $x + 9 = 15$.

Transpose $+9$ to the right:

Example 2: Solve $4x + 7 = 31$.

Transpose $+7$: $4x = 31 - 7 = 24$.
Transpose $\times 4$: $x = \frac{24}{4} = 6$.

Example 3: Solve $\frac{x}{3} + 2 = 8$.

Transpose $+2$: $\frac{x}{3} = 6$.
Transpose $\div 3$: $x = 6 \times 3 = 18$.

Verify: $\frac{18}{3} + 2 = 6 + 2 = 8$. Correct!

Example 4: Solve $2x - 5 = 3$.

Transpose $-5$: $2x = 3 + 5 = 8$.
Transpose $\times 2$: $x = \frac{8}{2} = 4$.

Transposing is faster once you're comfortable with it, but make sure you understand the balancing method first!

Word problems test your ability to form AND solve equations. Follow this process every time:

1. Read the problem carefully.
2. Identify the unknown and assign it a variable.
3. Form the equation from the given information.
4. Solve the equation.
5. Check that your answer makes sense.

Problem 1: Number Problem
The sum of three consecutive numbers is $84$. Find them.

Let the numbers be $x$, $x+1$, $x+2$.

The numbers are $27, 28, 29$. Check: $27 + 28 + 29 = 84$. Correct!

Problem 2: Age Problem
Rahul is $4$ years older than Priya. The sum of their ages is $28$. Find their ages.

Let Priya's age $= x$. Then Rahul's age $= x + 4$.

Priya is $12$, Rahul is $16$. Check: $12 + 16 = 28$. Correct!

Problem 3: Money Problem
A pen costs Rs. $5$ more than a pencil. If $3$ pens and $2$ pencils cost Rs. $55$, find the cost of each.

Let pencil cost $= x$. Pen cost $= x + 5$.

Pencil costs Rs. $8$, pen costs Rs. $13$. Check: $3(13) + 2(8) = 39 + 16 = 55$. Correct!

Here's everything you need to remember:

• An equation has an equals sign; an expression does not.
  - The solution is the value of the variable that makes the equation true.
  - Balancing method: Do the same operation to both sides.
  - Transposing: Move terms across the equals sign by flipping their sign/operation.
  - For word problems: read, assign variable, form equation, solve, verify.
  - Always check your answer by substituting back.

Ready to practice? Head to SparkEd and solve interactive simple equation problems with instant feedback. Every problem you solve makes you a little bit stronger!