What is a Linear Equation with Variables on Both Sides?

S3-SA1-0275

Grade Level:

Class 6

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

Definition

What is it?

A linear equation with variables on both sides is like a balanced seesaw where unknown values (variables) are present on both sides of the equals sign. Your goal is to find the value of this unknown that makes both sides equal. It's called 'linear' because when you plot it, it forms a straight line.

Simple Example

Quick Example

Imagine your friend Rohan has some cricket cards, and you have some. Rohan says, 'If I double my cards and add 5, I'll have the same number as if you triple your cards and subtract 2.' This can be written as 2x + 5 = 3x - 2, where 'x' is the unknown number of cards.

Worked Example

Step-by-Step

Let's solve: 5x + 3 = 2x + 12

Step 1: Gather all terms with 'x' on one side. Subtract 2x from both sides.
5x + 3 - 2x = 2x + 12 - 2x
3x + 3 = 12

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Step 2: Gather all constant numbers on the other side. Subtract 3 from both sides.
3x + 3 - 3 = 12 - 3
3x = 9

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Step 3: Isolate 'x' by dividing both sides by the number next to 'x'. Divide both sides by 3.
3x / 3 = 9 / 3
x = 3

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Answer: x = 3

Why It Matters

Understanding these equations is crucial for many fields like Data Science and Engineering, where you need to solve for unknown values in real-world problems. Engineers use them to design bridges, and economists use them to predict market trends, helping them make important decisions for our country's growth.

Common Mistakes

MISTAKE: Forgetting to change the sign of a term when moving it to the other side of the equals sign. For example, moving +2x as +2x. | CORRECTION: When you move a term across the equals sign, always change its operation (plus becomes minus, minus becomes plus). So, +2x becomes -2x.

MISTAKE: Combining terms that are not 'like terms' (e.g., adding 3x and 5). | CORRECTION: Only add or subtract terms that have the same variable and power (like 3x and 2x) or terms that are just numbers (like 5 and 10).

MISTAKE: Incorrectly dividing or multiplying only one side of the equation. | CORRECTION: Whatever operation you perform on one side of the equals sign (add, subtract, multiply, divide), you MUST perform the exact same operation on the other side to keep the equation balanced.

Practice Questions

Try It Yourself

QUESTION: Solve for y: 4y + 7 = y + 16 | ANSWER: y = 3

QUESTION: The cost of 3 samosas and 10 rupees for a drink is equal to the cost of 5 samosas minus 4 rupees. Find the cost of one samosa. (Let 's' be the cost of one samosa). | ANSWER: s = 7

QUESTION: If 2 times a number, increased by 15, is equal to 5 times the same number, decreased by 9, what is the number? | ANSWER: The number is 8.

MCQ

Quick Quiz

Which of these equations has variables on both sides?

3x + 5 = 11

7 = 2y - 4

4z + 1 = z - 8

x/2 = 6

The Correct Answer Is:

Option C, 4z + 1 = z - 8, has the variable 'z' on both the left and right sides of the equals sign. The other options only have the variable on one side.

Real World Connection

In the Real World

Imagine you're comparing mobile data plans. One plan offers '3GB per month plus 50 rupees extra for calls' and another offers '2GB per month plus 100 rupees extra for calls' for the same total bill. You could use a linear equation with variables on both sides to figure out how many extra GBs (the variable) would make the total cost equal for both plans. This helps telecom companies set fair prices.

Key Vocabulary

Key Terms

VARIABLE: A letter (like x, y, z) that represents an unknown number. | EQUATION: A mathematical statement that shows two expressions are equal using an equals sign (=). | LINEAR: Means that when graphed, the equation forms a straight line. | CONSTANT: A number that has a fixed value and does not change.

What's Next

What to Learn Next

Great job learning about linear equations with variables on both sides! Next, you can explore 'Solving Linear Equations with Brackets' or 'Linear Equations with Fractions'. These will build on what you've learned and help you solve even more complex problems, making you a math whiz!