What is Substitution of One Variable? | Simple Explanation for Class 10

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What is the Substitution of One Variable?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

Substitution of one variable means replacing a variable in an equation or expression with its known value or an equivalent expression. It's like swapping out one thing for another that means the same, to simplify or solve a problem.

Simple Example

Quick Example

Imagine you know the price of one samosa (let's say 's') is Rs. 15. If your friend asks you to calculate the cost of 3 samosas and a chai, and the chai costs Rs. 10, you can substitute 's' with 15. So, the cost becomes (3 * 15) + 10 = 45 + 10 = Rs. 55.

Worked Example

Step-by-Step

PROBLEM: Solve the equations: Equation 1: y = x + 3 | Equation 2: 2x + y = 9
---STEP 1: Identify the variable to substitute. From Equation 1, we know y is equal to (x + 3).
---STEP 2: Substitute the expression for 'y' from Equation 1 into Equation 2. So, 2x + (x + 3) = 9.
---STEP 3: Simplify the new equation. Combine the 'x' terms: 3x + 3 = 9.
---STEP 4: Isolate the 'x' term. Subtract 3 from both sides: 3x = 9 - 3, which means 3x = 6.
---STEP 5: Solve for 'x'. Divide both sides by 3: x = 6 / 3, so x = 2.
---STEP 6: Now that you have the value of 'x', substitute it back into either original equation to find 'y'. Let's use Equation 1: y = x + 3.
---STEP 7: Substitute x = 2 into y = x + 3: y = 2 + 3, so y = 5.
---ANSWER: The solution is x = 2 and y = 5.

Why It Matters

This technique is super important for solving complex problems in science and technology. Engineers use it to design bridges, AI developers use it to train smart systems, and even doctors use it to calculate medicine dosages. It helps simplify big problems into smaller, solvable parts.

Common Mistakes

MISTAKE: Forgetting to substitute into ALL parts of the expression or equation, especially when there are multiple terms with the variable. | CORRECTION: Double-check that every instance of the variable you are substituting has been replaced with its new value or expression.

MISTAKE: Making calculation errors after substitution, like incorrect addition, subtraction, or sign changes. | CORRECTION: Always use parentheses when substituting an expression to avoid sign errors, and carefully perform each arithmetic operation step-by-step.

MISTAKE: Not solving for both variables in a system of equations; stopping after finding the first variable. | CORRECTION: Remember that for a system of two variables, you need to find the value of both variables (e.g., both x and y) by substituting back your first answer.

Practice Questions

Try It Yourself

QUESTION: If a = 5, what is the value of the expression 3a + 7? | ANSWER: 22

QUESTION: Solve for x and y: Equation 1: x = 2y | Equation 2: 3x - y = 10 | ANSWER: x = 4, y = 2

QUESTION: The total cost of 2 notebooks (N) and 3 pens (P) is Rs. 110. If one notebook costs Rs. 5 more than one pen (N = P + 5), find the cost of one pen. | ANSWER: Rs. 20

MCQ

Quick Quiz

If x = y - 4 and 2x + y = 7, what is the value of y?

The Correct Answer Is:

Substitute x = y - 4 into 2x + y = 7 to get 2(y - 4) + y = 7. This simplifies to 2y - 8 + y = 7, then 3y - 8 = 7, so 3y = 15, which means y = 5.

Real World Connection

In the Real World

When you use a navigation app like Google Maps or Ola, it calculates the estimated time of arrival. This often involves substituting variables like average speed (based on traffic) into distance-time formulas to give you a real-time update. Even calculating your mobile data usage involves substituting your usage into a plan's formula to see your remaining data.

Key Vocabulary

Key Terms

VARIABLE: A symbol (like x or y) that represents a quantity that can change | EXPRESSION: A combination of numbers, variables, and operation symbols, but no equality sign | EQUATION: A statement that two expressions are equal, shown with an equality sign (=) | SUBSTITUTE: To replace something with another equivalent thing

What's Next

What to Learn Next

Great job understanding substitution! Next, you should explore 'Solving Systems of Linear Equations by Substitution'. This builds directly on what you've learned, showing you how to solve more complex problems with multiple equations and variables, a key skill for higher mathematics.