What is an Equation with Two Variables? | Simple Explanation for Class 7

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What is an Equation with Two Variables (General Form)?

Grade Level:

Class 7

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

Definition

What is it?

An equation with two variables is a mathematical statement that shows two expressions are equal, where both expressions contain two different unknown values. These unknown values are represented by letters, usually 'x' and 'y', and they can change. The general form of such an equation is ax + by + c = 0, where 'a', 'b', and 'c' are fixed numbers (constants) and 'a' and 'b' are not zero.

Simple Example

Quick Example

Imagine you are buying samosas and jalebis from your local shop. If each samosa costs Rs. 10 and each jalebi costs Rs. 20, and you spend a total of Rs. 100, we can write this as an equation. Let 'x' be the number of samosas and 'y' be the number of jalebis. The equation would be 10x + 20y = 100. Here, 'x' and 'y' are the two variables.

Worked Example

Step-by-Step

Let's convert the statement 'The cost of 2 pens and 3 notebooks is Rs. 70' into an equation with two variables in its general form.

Step 1: Identify the unknown quantities. The number of pens is known (2), and the number of notebooks is known (3). What's unknown is the cost of one pen and the cost of one notebook.
---Step 2: Assign variables to these unknown quantities. Let the cost of one pen be 'x' rupees and the cost of one notebook be 'y' rupees.
---Step 3: Write the expression for the total cost of pens. If one pen costs 'x', then 2 pens cost 2 * x, or 2x.
---Step 4: Write the expression for the total cost of notebooks. If one notebook costs 'y', then 3 notebooks cost 3 * y, or 3y.
---Step 5: Form the equation using the total given cost. The total cost is Rs. 70, so 2x + 3y = 70.
---Step 6: Convert to the general form (ax + by + c = 0). To do this, move the constant term (70) to the left side of the equation. This makes it 2x + 3y - 70 = 0.
---Step 7: Identify 'a', 'b', and 'c'. Here, a = 2, b = 3, and c = -70.

Answer: The equation in general form is 2x + 3y - 70 = 0.

Why It Matters

Understanding equations with two variables is crucial for fields like Data Science and AI, where you often model relationships between different factors. Engineers use them to design structures and systems, and economists use them to predict market trends. They are the building blocks for solving complex real-world problems.

Common Mistakes

MISTAKE: Not moving all terms to one side for the general form (e.g., leaving it as 2x + 3y = 70). | CORRECTION: Remember the general form is ax + by + c = 0. So, all terms, including the constant, must be on one side, with the other side equal to zero.

MISTAKE: Confusing the coefficients 'a' and 'b' with the variables 'x' and 'y'. (e.g., thinking 'a' is a variable). | CORRECTION: 'x' and 'y' are the variables (unknowns). 'a', 'b', and 'c' are constants (fixed numbers) that multiply the variables or stand alone.

MISTAKE: Forgetting the sign of 'c' when writing the general form (e.g., writing 2x + 3y + 70 = 0 instead of 2x + 3y - 70 = 0). | CORRECTION: When moving a term from one side of the equals sign to the other, always change its sign. So, if +70 moves from right to left, it becomes -70.

Practice Questions

Try It Yourself

QUESTION: Write the equation for 'The sum of two numbers is 15' using x and y as variables. | ANSWER: x + y = 15

QUESTION: Convert the statement 'The cost of a movie ticket (x) is Rs. 120 more than the cost of a popcorn tub (y)' into an equation in general form. | ANSWER: x - y - 120 = 0

QUESTION: A mobile phone plan charges a fixed monthly fee of Rs. 200 plus Rs. 5 for every GB of data used. If the total bill for a month is Rs. 350, and 'x' is the data used in GB, and 'y' is the fixed monthly fee, write the equation in general form. | ANSWER: 5x + y - 350 = 0 (Note: Here, y is the fixed fee, which is 200, but the question asks for the general form based on total bill, so 5x + 200 = 350 leads to 5x + 200 - 350 = 0, which is 5x - 150 = 0. But if y is the variable for the fixed fee, then 5x + y = 350. Let's re-evaluate for clarity.) Let's assume 'y' represents the total bill. This question is tricky. Let's rephrase for Class 7. QUESTION: A shop sells two types of sweets: Laddoo (x rupees per piece) and Barfi (y rupees per piece). If you buy 5 Laddoos and 4 Barfis for a total of Rs. 220, write the equation in general form. | ANSWER: 5x + 4y - 220 = 0

MCQ

Quick Quiz

Which of the following is an equation with two variables in its general form?

3x + 5 = 0

4y = 12

2x - 7y + 10 = 0

x + y + z = 15

The Correct Answer Is:

Option C (2x - 7y + 10 = 0) has two variables ('x' and 'y') and is written in the ax + by + c = 0 format. Option A has only one variable. Option B has only one variable. Option D has three variables.

Real World Connection

In the Real World

When you use a food delivery app like Swiggy or Zomato, the total bill depends on the cost of the food items and the delivery charges. If 'x' is the total cost of food and 'y' is the delivery charge, the total amount you pay could be represented as an equation like x + y = Total Bill. Similarly, ISRO scientists use equations with multiple variables to calculate rocket trajectories and satellite orbits.

Key Vocabulary

Key Terms

VARIABLE: A letter (like x or y) that represents an unknown number that can change. | CONSTANT: A fixed numerical value that does not change. | COEFFICIENT: The number multiplied by a variable (e.g., 'a' in 'ax'). | EQUATION: A mathematical statement showing that two expressions are equal. | GENERAL FORM: A standard way to write an equation, like ax + by + c = 0.

What's Next

What to Learn Next

Great job learning about equations with two variables! Next, you should explore how to find solutions for these equations, and then move on to graphing them. This will help you visualize the relationship between the variables and understand how they change together.