What is the Substitution Method? | Simple Explanation for Class 10

S6-SA1-0007

What is the Substitution Method for Linear Equations?

Grade Level:

Class 10

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

Definition

What is it?

The Substitution Method is a way to solve a system of two linear equations with two variables. It involves expressing one variable in terms of the other from one equation and then substituting this expression into the second equation to find the value of one variable.

Simple Example

Quick Example

Imagine you bought 2 samosas and 1 chai for Rs. 50. Your friend bought 1 samosa and 1 chai for Rs. 30. Using the substitution method, we can figure out the individual price of one samosa and one chai.

Worked Example

Step-by-Step

Let's solve the system of equations:
1) x + y = 10
2) 2x - y = 5

Step 1: Express one variable in terms of the other from one equation. From equation (1), we can write y = 10 - x.
---Step 2: Substitute this expression for 'y' into the second equation (2). So, 2x - (10 - x) = 5.
---Step 3: Simplify and solve for 'x'. 2x - 10 + x = 5 => 3x - 10 = 5 => 3x = 15 => x = 5.
---Step 4: Substitute the value of 'x' back into the expression for 'y' from Step 1. y = 10 - 5 => y = 5.
---Answer: So, x = 5 and y = 5.

Why It Matters

This method is super useful in fields like engineering to design circuits or in physics to calculate forces. Even AI/ML engineers use similar logic to optimize algorithms. It's a foundational skill for many STEM careers like data scientists or researchers.

Common Mistakes

MISTAKE: Substituting the expression back into the *same* equation it came from. | CORRECTION: Always substitute the expression into the *other* equation to form a new equation with only one variable.

MISTAKE: Making calculation errors when simplifying the substituted equation, especially with negative signs. | CORRECTION: Be very careful with arithmetic, especially when distributing negative signs, e.g., -(a - b) becomes -a + b.

MISTAKE: Forgetting to find the value of the second variable after finding the first. | CORRECTION: Once you find one variable, always substitute its value back into one of the original equations (or the derived expression) to find the other variable.

Practice Questions

Try It Yourself

QUESTION: Solve using substitution: x + y = 7 and x - y = 3. | ANSWER: x = 5, y = 2

QUESTION: Solve using substitution: 2x + y = 8 and x - 2y = -1. | ANSWER: x = 3, y = 2

QUESTION: The total cost of 3 pens and 2 notebooks is Rs. 100. If a notebook costs Rs. 10 more than a pen, find the cost of each. | ANSWER: Pen = Rs. 16, Notebook = Rs. 26

MCQ

Quick Quiz

Which of these is the first step in the substitution method?

Add the two equations together

Multiply both equations by a constant

Express one variable in terms of the other from one equation

Graph both equations to find the intersection point

The Correct Answer Is:

The first step in the substitution method is to isolate one variable in one of the equations. Options A and B are steps for the elimination method, and D is for the graphical method.

Real World Connection

In the Real World

Think about planning your day! If you have a fixed amount of time (say, 5 hours) and two tasks (studying and playing), and you know studying takes twice as long as playing, you can use a similar logic to figure out how much time to spend on each. Or, a small business owner might use this to calculate how many units of two different products to sell to reach a profit target, given their individual costs and selling prices.

Key Vocabulary

Key Terms

LINEAR EQUATION: An equation where the highest power of the variable is 1, resulting in a straight line graph. | SYSTEM OF EQUATIONS: A set of two or more equations with the same variables that need to be solved simultaneously. | VARIABLE: A symbol (usually a letter like x or y) representing an unknown value. | SUBSTITUTE: To replace a variable with an expression or numerical value.

What's Next

What to Learn Next

Great job mastering substitution! Next, you should explore the 'Elimination Method' for solving linear equations. It's another powerful technique that sometimes makes solving even quicker, and understanding both will make you a pro at solving systems of equations!