Algebraic Expressions Ex 12.2: CBSE Class 7 Maths NCERT Guide

Welcome, students! In this chapter, we are exploring the world of Algebraic Expressions. Think of them as mathematical sentences with variables (like x, y), numbers, and operations (+, -, ×, ÷). In the previous exercise, you learned how to identify terms, factors, and coefficients. Now, in Exercise 12.2, we'll learn how to perform operations on these expressions. Specifically, we will master adding and subtracting them.

Why is this important? Just like you can add 5 apples and 3 apples, you can add algebraic terms that are similar. This skill is a building block for solving almost all future math problems in algebra. By the end of this guide, you will be able to confidently simplify complex expressions by combining like terms and perform addition and subtraction without any confusion. Let's get started with your YoLearn AI Tutor!

The Golden Rule: Like and Unlike Terms

Before we can add or subtract, we must understand the most important rule in algebra: You can only add or subtract 'like terms'.

So, what are like terms?
Like terms are terms that have the exact same algebraic factors. This means they must have the same variables, and each variable must have the same power (exponent).

  • Example of Like Terms: 7x, -2x, 0.5x. Here, the variable is x in all terms, and its power is 1. The numerical coefficients (7, -2, 0.5) can be different.
  • Another example: 5x²y, 9x²y, -x²y. The variables are x² and y, and they are the same in all three terms.

What are unlike terms then? These are terms that do not have the same algebraic factors.

  • Example of Unlike Terms: 7x and 7y. They have different variables.
  • Another example: 3a² and 8a. They have the same variable a, but the powers are different (2 and 1).
  • And another: 4mn and 5mp. They do not have the exact same variables (mn vs mp).

Think of it like fruits. You can add 3 apples and 4 apples to get 7 apples. But you can't add 3 apples and 4 bananas to get '7 apple-bananas'. In algebra, 3x + 4x = 7x, but 3x + 4y cannot be simplified further. It just stays 3x + 4y.

How to Add Algebraic Expressions

  1. Step 1: Write Down the Expressions — Place the expressions you need to add next to each other. It's helpful to keep each expression in brackets initially. For example, to add (2x + 3y) and (4x - y), you write: (2x + 3y) + (4x - y).
  2. Step 2: Group Like Terms Together — Rearrange the terms so that all the like terms are next to each other. Use the sign that is in front of each term. From our example: 2x + 4x + 3y - y. We've grouped the 'x' terms and the 'y' terms.
  3. Step 3: Add the Coefficients of Like Terms — Now, perform the addition on the numerical coefficients of each group of like terms. Remember, if a variable has no number, its coefficient is 1 or -1. So, (2 + 4)x and (3 - 1)y.
  4. Step 4: Write the Final Simplified Expression — Combine the results from the previous step to get your final answer. 6x + 2y. Since 6x and 2y are unlike terms, we cannot simplify this any further.

Worked Examples: Subtracting Algebraic Expressions

  • Example 1: Subtract (4a - 2b) from (9a + 5b) Step 1: Write the expression correctly. 'From' means the second expression comes first. So, we write: (9a + 5b) - (4a - 2b). Step 2: This is the most crucial step in subtraction! Change the sign of every term inside the second bracket. The +4a becomes -4a and the -2b becomes +2b. The expression becomes: 9a + 5b - 4a + 2b. Step 3: Group the like terms: 9a - 4a + 5b + 2b. Step 4: Combine the coefficients: (9 - 4)a + (5 + 2)b. Final Answer: 5a + 7b.
  • Example 2: Subtract (x² - 2xy + 3y²) from (5x² + 4xy - y²) Step 1: Write the expression: (5x² + 4xy - y²) - (x² - 2xy + 3y²). Step 2: Open the brackets and reverse the signs of the second expression: 5x² + 4xy - y² - x² + 2xy - 3y². Step 3: Group all the like terms together carefully: (5x² - x²) + (4xy + 2xy) + (-y² - 3y²). Step 4: Add or subtract the coefficients for each group: (5 - 1)x² + (4 + 2)xy + (-1 - 3)y². Final Answer: 4x² + 6xy - 4y².

Common Mistakes to Avoid

  • Combining Unlike Terms: A very common error is to add unlike terms, for example, writing 5x + 2y = 7xy. This is incorrect! Remember, apples and oranges can't be added. 5x + 2y is the simplest form.
  • Forgetting to Change Signs in Subtraction: When subtracting (a - b), students often only change the sign of the first term (-a) and forget the second one. The correct way is -a + b. Always change the sign of every term in the expression being subtracted.
  • Ignoring Invisible Coefficients: Terms like x or -y have coefficients of 1 and -1 respectively. Don't forget to include these when you are adding or subtracting.
  • Column Method Errors: If using the column method for subtraction, remember to change the signs of all terms in the lower expression before you start adding vertically.

Practice Questions with Solutions

  • Q: Simplify by combining like terms: 3a - 2b - ab - (a - b + ab) + 3ab + b - a A: Step 1: First, let's open the brackets. Remember to change the signs of the terms inside the bracket being subtracted. 3a - 2b - ab - a + b - ab + 3ab + b - a Step 2: Now, group the like terms together. Group all 'a' terms, 'b' terms, and 'ab' terms. (3a - a - a) + (-2b + b + b) + (-ab - ab + 3ab) Step 3: Combine the coefficients of each group. For 'a': (3 - 1 - 1)a = 1a = a For 'b': (-2 + 1 + 1)b = 0b = 0 For 'ab': (-1 - 1 + 3)ab = 1ab = ab Final answer: a + ab
  • Q: Add the following expressions: 7x + 5y - 3z, 2y + z - 4x, and x - 3y - 2z A: Step 1: Write the expressions to be added together. We can arrange them in columns for clarity, aligning the like terms. 7x + 5y - 3z -4x + 2y + z x - 3y - 2z ------------------ Step 2: Add the coefficients of the 'x' terms: 7 - 4 + 1 = 4 Step 3: Add the coefficients of the 'y' terms: 5 + 2 - 3 = 4 Step 4: Add the coefficients of the 'z' terms: -3 + 1 - 2 = -4 Final answer: 4x + 4y - 4z
  • Q: What should be subtracted from 2a + 8b + 10 to get -3a + 7b + 16? A: Step 1: This is a word problem. Let the expression to be subtracted be 'X'. The problem can be written as: (2a + 8b + 10) - X = -3a + 7b + 16. Step 2: To find X, we can rearrange the equation: X = (2a + 8b + 10) - (-3a + 7b + 16). Step 3: Now, we perform the subtraction. Open the second bracket and change the sign of each term inside. X = 2a + 8b + 10 + 3a - 7b - 16 Step 4: Group the like terms. X = (2a + 3a) + (8b - 7b) + (10 - 16) Step 5: Combine the coefficients. X = 5a + 1b - 6 Final answer: 5a + b - 6 should be subtracted.
  • Q: From the sum of 3x - y + 11 and -y - 11, subtract 3x - y - 11. A: Step 1: First, find the sum of (3x - y + 11) and (-y - 11). Sum = 3x - y + 11 - y - 11 Group like terms: 3x + (-y - y) + (11 - 11) Sum = 3x - 2y + 0 = 3x - 2y Step 2: Now, subtract 3x - y - 11 from this sum. (3x - 2y) - (3x - y - 11) Step 3: Open the brackets and reverse the signs of the second expression. 3x - 2y - 3x + y + 11 Step 4: Group the like terms. (3x - 3x) + (-2y + y) + 11 0x - 1y + 11 Final answer: -y + 11

Frequently Asked Questions

What are 'like terms' in algebra?

Like terms are terms that have the exact same variables raised to the exact same powers. For example, `5x²` and `-3x²` are like terms, but `5x²` and `5x` are not because the powers of 'x' are different.

Why can't we add or subtract unlike terms?

You can't combine unlike terms because they represent different kinds of quantities. Think of it like trying to add 4 apples (`4a`) and 3 bananas (`3b`). You can't simplify `4a + 3b` into a single term, just as you can't say you have '7 apple-bananas'.

What is the most common mistake in subtracting algebraic expressions?

The most common mistake is forgetting to change the sign of *every* term in the expression being subtracted. When you have `A - (B - C)`, it should become `A - B + C`, not `A - B - C`.

Is there a trick to remember for subtraction?

Yes! A good trick is to 'add the opposite'. Instead of subtracting an expression, you add its opposite version. To get the opposite, just flip the sign of every single term inside the expression's brackets.