What is Solving Equations with Variables on Both Sides?

S3-SA1-0690

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Solving equations with variables on both sides means finding the value of the unknown variable when it appears on both the left and right sides of the equals sign. We use algebraic operations to isolate the variable on one side and constant numbers on the other.

Simple Example

Quick Example

Imagine you and your friend, Rohan, are saving money. You have Rs 50 and save Rs 10 every day. Rohan has Rs 20 and saves Rs 15 every day. To find out when you both will have the same amount of money, we can set up an equation: 50 + 10x = 20 + 15x, where 'x' is the number of days.

Worked Example

Step-by-Step

Let's solve: 3x + 7 = x + 15

1. Identify the variable terms (3x and x) and constant terms (7 and 15).
2. Move all variable terms to one side. Subtract 'x' from both sides:
3x - x + 7 = x - x + 15
2x + 7 = 15
3. Move all constant terms to the other side. Subtract '7' from both sides:
2x + 7 - 7 = 15 - 7
2x = 8
4. Isolate the variable 'x'. Divide both sides by '2':
2x / 2 = 8 / 2
x = 4

Answer: x = 4

Why It Matters

Understanding this helps in many fields like predicting stock prices in Economics, optimizing delivery routes in Logistics, and even designing better algorithms in Computer Science. Engineers use this to balance forces in structures, and Data Scientists use it to build predictive models.

Common Mistakes

MISTAKE: Adding or subtracting a term only from one side of the equation. | CORRECTION: Always perform the same operation (add, subtract, multiply, divide) on BOTH sides of the equals sign to keep the equation balanced.

MISTAKE: Incorrectly changing the sign of a term when moving it to the other side of the equation. | CORRECTION: When you move a term from one side to the other, its sign changes. For example, moving '+5' becomes '-5' on the other side.

MISTAKE: Forgetting to combine like terms (e.g., adding 3x and 5 instead of 3x and 2x). | CORRECTION: Only combine terms that have the same variable and power (like 'x' with 'x') or are both constants (like '5' with '7').

Practice Questions

Try It Yourself

QUESTION: Solve for y: 5y - 8 = 2y + 1 | ANSWER: y = 3

QUESTION: Find the value of 'a': 4a + 10 = 7a - 5 | ANSWER: a = 5

QUESTION: If 2(x + 3) = 5x - 3, what is x? | ANSWER: x = 3

MCQ

Quick Quiz

What is the first step to solve 6x + 4 = 2x + 16?

Add 4 to both sides

Subtract 2x from both sides

Divide both sides by 6

Subtract 16 from both sides

The Correct Answer Is:

The first step is usually to gather all variable terms on one side. Subtracting 2x from both sides moves the variable term from the right to the left.

Real World Connection

In the Real World

Imagine a mobile network provider like Jio or Airtel. They might have different plans: Plan A costs Rs 200 + Rs 5 per GB, and Plan B costs Rs 100 + Rs 10 per GB. To find out for how many GBs the cost will be the same, you'd set up an equation like 200 + 5x = 100 + 10x. Solving this helps them understand customer choices and pricing strategies.

Key Vocabulary

Key Terms

VARIABLE: A letter representing an unknown number, like 'x' or 'y' | EQUATION: A mathematical statement showing two expressions are equal, using an '=' sign | CONSTANT: A fixed numerical value in an equation, like '7' or '15' | TERM: A single number, variable, or product of numbers and variables within an expression

What's Next

What to Learn Next

Great job mastering this! Next, you can explore 'Solving Equations with Brackets' or 'Solving Equations with Fractions'. These build upon the skills you've learned here, making equations a little more complex but still solvable using similar balancing techniques.