Simple Equations: Solving Equations (Ex 4.2) for Class 7

Welcome, students! In the previous exercise, you learned how to set up simple equations from statements. Now, we take the next exciting step: finding the solution! Exercise 4.2 is all about solving these equations to find the value of the unknown variable. Think of an equation as a perfectly balanced weighing scale. The equals sign (=) is the center point. To keep it balanced, whatever you do to one side, you must do to the other. This simple idea is the key to unlocking every problem in this exercise. In this guide, we'll walk you through the method of isolating the variable by using inverse operations. By the end, you'll be able to solve equations like x + 5 = 10 or 4y = 24 with ease and confidence.

The Balancing Act: How to Solve Simple Equations

The fundamental rule for solving any equation is to keep it balanced. Imagine the two sides of an equation on a weighing scale. If you add a weight to one side, you must add the same weight to the other to keep it level. The same logic applies to equations. Our goal is to find what the variable (like 'x', 'y', or 'p') equals. To do this, we need to get it all by itself on one side of the equals sign. This is called 'isolating the variable'. We achieve this by using 'inverse operations' – the opposite action that will undo what's being done to the variable.

  • To undo addition, we subtract from both sides.
  • To undo subtraction, we add to both sides.
  • To undo multiplication, we divide both sides.
  • To undo division, we multiply both sides.

By systematically applying these inverse operations, you can peel away the numbers surrounding the variable until it stands alone, revealing its value.

Step-by-Step Method for Solving Equations

  1. Step 1: Identify the Goal and the Operation — Look at the equation and identify the variable you need to solve for. See what operation is being performed on the variable. For example, in y + 6 = 10, the variable is 'y' and 6 is being added to it.
  2. Step 2: Choose the Inverse Operation — To isolate the variable, you need to 'undo' the operation. The inverse of addition is subtraction. The inverse of multiplication is division, and so on. For y + 6 = 10, the inverse operation is subtracting 6.
  3. Step 3: Apply the Inverse Operation to Both Sides — This is the most crucial step. To keep the equation balanced, perform the inverse operation on both sides of the equals sign. For y + 6 = 10, we subtract 6 from both sides: y + 6 - 6 = 10 - 6.
  4. Step 4: Simplify to Find the Solution — Perform the calculations on both sides. On the variable's side, the operation and its inverse will cancel each other out. On the other side, you'll get a number. So, y = 4. You have found the solution!

Worked Examples: Solving for the Variable

  • Example 1 (Using Subtraction): Solve x + 5 = 12. Step 1: 5 is being added to x. The inverse is to subtract 5. Step 2: Subtract 5 from both sides: x + 5 - 5 = 12 - 5. Step 3: Simplify: x = 7.
  • Example 2 (Using Addition): Solve p - 4 = 3. Step 1: 4 is being subtracted from p. The inverse is to add 4. Step 2: Add 4 to both sides: p - 4 + 4 = 3 + 4. Step 3: Simplify: p = 7.
  • Example 3 (Using Division): Solve 3n = 21. Step 1: n is being multiplied by 3. The inverse is to divide by 3. Step 2: Divide both sides by 3: 3n / 3 = 21 / 3. Step 3: Simplify: n = 7.
  • Example 4 (Using Multiplication): Solve m / 5 = 4. Step 1: m is being divided by 5. The inverse is to multiply by 5. Step 2: Multiply both sides by 5: (m / 5) 5 = 4 5. Step 3: Simplify: m = 20.

Key Points to Remember

  • The Golden Rule: Always perform the same operation on BOTH sides of the equals sign. Forgetting to do it on one side is the most common mistake!
  • Signs Matter: Pay close attention to positive (+) and negative (-) signs. When you use an inverse operation, you are cancelling out the original term. The inverse of adding 3 is subtracting 3; the inverse of subtracting 3 is adding 3.
  • Check Your Answer: After you find a value for the variable, put it back into the original equation to see if it works. For example, if you found x = 7 for x + 5 = 12, check if 7 + 5 really equals 12. Yes, it does! This is a great way to catch mistakes.

Practice Questions with Solutions

  • Q: Give the first step you will use to separate the variable and then solve the equation: y + 9 = 15 A: Step 1: The first step is to separate the variable 'y'. Since 9 is added to 'y', we will subtract 9 from both sides of the equation. Step 2: y + 9 - 9 = 15 - 9 Step 3: Simplifying the equation, we get y = 6. Final answer: First step is to subtract 9 from both sides. The solution is y = 6.
  • Q: Give the first step you will use to separate the variable and then solve the equation: a - 7 = -2 A: Step 1: To separate the variable 'a', we need to undo the subtraction of 7. The inverse operation is to add 7 to both sides. Step 2: a - 7 + 7 = -2 + 7 Step 3: Simplifying the equation, we get a = 5. Final answer: First step is to add 7 to both sides. The solution is a = 5.
  • Q: Solve the equation: 8x = 56 A: Step 1: The variable 'x' is multiplied by 8. To isolate 'x', we must perform the inverse operation, which is to divide by 8. Step 2: Divide both sides by 8: 8x / 8 = 56 / 8 Step 3: Simplifying the equation, we get x = 7. Final answer: x = 7.
  • Q: Solve the equation: b / 4 = 12 A: Step 1: The variable 'b' is divided by 4. To isolate 'b', we perform the inverse operation, which is to multiply by 4. Step 2: Multiply both sides by 4: (b / 4) 4 = 12 4 Step 3: Simplifying the equation, we get b = 48. Final answer: b = 48.

Frequently Asked Questions

What is the main goal when solving a simple equation?

The main goal is to find the value of the unknown variable. You do this by isolating the variable (like 'x' or 'y') on one side of the equals sign using inverse operations.

What does it mean to 'balance' an equation?

Balancing an equation means whatever operation (add, subtract, multiply, divide) you do on one side, you must do the exact same thing on the other side. This ensures the equality remains true, just like keeping a weighing scale balanced.

How do I solve an equation like 2x = 10?

In `2x = 10`, the variable 'x' is being multiplied by 2. To solve it, you do the inverse operation, which is division. Divide both sides by 2 to get `x = 5`.