CBSE Class 6 Maths: Algebra Exercise 11.5 - Mastering Equations

Welcome, young mathematicians, to Exercise 11.5 of your Algebra chapter! This section is all about diving deeper into the exciting world of equations. You've already learned about variables, which are like secret codes for unknown numbers. Now, we're going to use these variables to build and solve equations, which are like puzzles where you need to find the value of the secret code.

Understanding equations is a super important skill. It helps you solve problems not just in maths, but in everyday life too – like figuring out how many chocolates you can share equally or calculating how much money you need for a new toy. In this exercise, we will focus on understanding what makes an equation true, how to check if a value is a solution, and practice solving simple equations. Get ready to put on your detective hat and uncover the mysteries of algebra!

What is an Equation?

An equation in mathematics is a statement that shows two mathematical expressions are equal. Think of it like a perfectly balanced weighing scale: whatever is on the left side must be exactly equal in value to whatever is on the right side. If the two sides are not equal, it's not an equation, or it's an equation that is currently false.

Equations always contain an equality sign (=). For example, in x + 5 = 12, the left side (x + 5) must have the same value as the right side (12). Here, 'x' is a variable, which is a letter used to represent an unknown number. Our goal when solving an equation is to find the value of this variable that makes the equation true. If x was 6, then 6 + 5 = 11, which is not equal to 12, so x = 6 is not a solution. But if x was 7, then 7 + 5 = 12, which IS equal to 12. So, x = 7 is the solution to this equation. This simple idea forms the foundation of all algebraic problem-solving!

Identifying the Solution of an Equation

  1. Understand the Goal — The 'solution' to an equation is the specific value of the variable that makes the equation true. It's the number that, when substituted in place of the variable, makes both sides of the '=' sign equal.
  2. Substitute the Given Value — When you are given a value and asked to check if it's a solution, simply replace the variable in the equation with that given number. Be careful with signs (positive/negative) and operations (addition, subtraction, multiplication, division).
  3. Calculate Both Sides — After substitution, calculate the value of the expression on the left side of the equation and the value of the expression on the right side separately.
  4. Compare the Results — If the calculated value of the Left Hand Side (LHS) is equal to the calculated value of the Right Hand Side (RHS), then the given value is indeed the solution. If they are not equal, then it is not the solution.
  5. Example: Is `y = 8` a solution to `y - 3 = 5`? — Step 1: The equation is y - 3 = 5. The given value is y = 8. Step 2: Substitute y = 8 into the equation: 8 - 3 = 5. Step 3: Calculate LHS: 8 - 3 = 5. RHS is 5. Step 4: Compare: 5 = 5. Yes, the LHS equals the RHS. So, y = 8 is a solution.

Solving Equations Using Trial and Error

  • Example 1: Solve m + 10 = 16 We need to find a value for 'm' that, when added to 10, gives 16. Try m = 5: 5 + 10 = 15. (Not 16) Try m = 6: 6 + 10 = 16. (This is 16!) So, the solution is m = 6.
  • Example 2: Solve t / 3 = 4 We need a number 't' that, when divided by 3, gives 4. Try t = 9: 9 / 3 = 3. (Not 4) Try t = 12: 12 / 3 = 4. (This is 4!) So, the solution is t = 12.
  • Example 3: Solve 4x = 20 Remember 4x means 4 x. We need a number 'x' that, when multiplied by 4, gives 20. Try x = 4: 4 4 = 16. (Not 20) Try x = 5: 4 * 5 = 20. (This is 20!) So, the solution is x = 5.

Exam Tip: Avoiding Common Mistakes

When working with equations in Exercise 11.5, students often make a few common errors. Firstly, always check your answer! After finding a value for the variable, substitute it back into the original equation to ensure both sides are equal. This simple step can catch most mistakes. Secondly, remember that multiplication is often hidden; 5p means 5 × p. Don't confuse it with 5 + p. Lastly, be careful with signs: x - 7 = 3 is different from x + 7 = 3. Take your time, read the question carefully, and verify each step.

Practice Questions with Solutions

  • Q: Is k = 9 a solution to the equation k + 7 = 15? A: Step 1: Substitute the value of k into the equation. 9 + 7 = 15 Step 2: Calculate the left side. 16 = 15 Step 3: Compare both sides. Since 16 is not equal to 15, k = 9 is NOT a solution. Final answer: No, k = 9 is not a solution.
  • Q: Is z = 4 a solution to the equation 3z = 12? A: Step 1: Substitute the value of z into the equation. 3 * 4 = 12 Step 2: Calculate the left side. 12 = 12 Step 3: Compare both sides. Since 12 is equal to 12, z = 4 IS a solution. Final answer: Yes, z = 4 is a solution.
  • Q: Find the value of 'a' that makes the equation a - 5 = 10 true, using trial and error. A: Step 1: Think of numbers that, when 5 is subtracted from them, result in 10. Step 2: Try a = 14: 14 - 5 = 9 (Not 10). Step 3: Try a = 15: 15 - 5 = 10 (This is 10!). Final answer: a = 15.
  • Q: Determine if p = 20 satisfies the equation p / 4 = 5. A: Step 1: Substitute the value of p into the equation. 20 / 4 = 5 Step 2: Calculate the left side. 5 = 5 Step 3: Compare both sides. Since 5 is equal to 5, p = 20 DOES satisfy the equation. Final answer: Yes, p = 20 satisfies the equation.

Frequently Asked Questions

What is a variable in an equation?

A variable is a letter (like x, y, a, m) used in an equation to represent an unknown number. Its value can change depending on the equation it's in, and finding that value is often the goal of solving an equation.

How do I know if a value is the 'solution' to an equation?

A value is a solution if, when you replace the variable with that value, both sides of the equation become equal. You substitute the number and then check if the Left Hand Side (LHS) matches the Right Hand Side (RHS).

Is trial and error the only way to solve equations in Class 6?

For simple equations in Class 6, trial and error is a common and effective method to understand the concept. As you progress, you'll learn more systematic ways, like balancing operations on both sides, to solve more complex equations.