NCERT Class 10 Maths: Linear Equations in Two Variables - Exercise 3.4 (Elimination Method)

Welcome, Class 10 students, to a crucial chapter in your mathematical journey: Linear Equations in Two Variables! In your previous classes, you've learned to solve linear equations with one variable. Now, we're stepping up to systems where you have two equations, each with two unknown variables, like 'x' and 'y'. Solving these systems means finding values for 'x' and 'y' that satisfy both equations simultaneously.

While there are various methods to solve these pairs of equations – graphical, substitution, and elimination – NCERT Exercise 3.4 specifically focuses on mastering the Elimination Method. This method is incredibly powerful and often more efficient than substitution for certain types of problems. By the end of this page, you will not only understand the logic behind elimination but also be able to apply it confidently to solve a wide range of problems, preparing you thoroughly for your CBSE board exams. Let's dive in and make these concepts crystal clear!

Understanding the Elimination Method

The Elimination Method is a systematic approach to solving a pair of linear equations by eliminating one of the variables. The core idea is to manipulate the given equations such that the coefficients of one variable become numerically equal (either with the same sign or opposite signs). Once the coefficients are equal, you can either add or subtract the equations to eliminate that variable, leaving you with a simple linear equation in a single variable, which you already know how to solve.

For example, if you have two equations like 2x + 3y = 10 and 2x - 5y = 2, notice that the coefficient of 'x' is already '2' in both. If you subtract the second equation from the first, the 'x' terms cancel out (2x - 2x = 0), and you're left with an equation involving only 'y' ((3y) - (-5y) = 10 - 2, which simplifies to 8y = 8). This makes solving for 'y' very straightforward. If the coefficients aren't initially equal, we multiply one or both equations by suitable non-zero constants to make them equal. The choice of which variable to eliminate often depends on convenience; pick the one whose coefficients are easier to make equal using the Least Common Multiple (LCM) concept. Remember, any operation (multiplication, addition, subtraction) must be applied to both sides of the equation to maintain balance.

Step-by-Step Guide: The Elimination Method

  1. Step 1: Write Down the Equations — Clearly write both linear equations, ensuring terms with the same variables are aligned. Label them as Equation (1) and Equation (2) for clarity.
  2. Step 2: Choose a Variable to Eliminate — Decide which variable (x or y) you want to eliminate. Look for coefficients that are already the same or can be easily made the same by multiplying one or both equations by a small integer.
  3. Step 3: Make Coefficients Equal — If the coefficients of your chosen variable are not already equal, multiply one or both equations by a suitable non-zero constant such that the absolute value of the coefficients of the chosen variable becomes the same. The goal is to find the LCM of the coefficients and make both coefficients equal to that LCM.
  4. Step 4: Add or Subtract the Equations — If the equal coefficients have opposite signs, add the two new equations. If they have the same sign, subtract one equation from the other. This step eliminates one variable, leaving a linear equation in one variable.
  5. Step 5: Solve for the Remaining Variable — Solve the resulting single-variable equation to find the value of that variable.
  6. Step 6: Substitute Back to Find the Other Variable — Substitute the value found in Step 5 into either the original Equation (1) or Equation (2). Solve this new equation to find the value of the second variable.
  7. Step 7: Verify Your Solution — Substitute both values (x and y) back into both original equations. If both equations are satisfied, your solution is correct.

Solved Examples Using Elimination

  • Example 1: Solve the following pair of linear equations by the elimination method: Equation (1): x + y = 5 Equation (2): 2x - 3y = 4 Solution: Step 1: We want to eliminate 'y'. Multiply Equation (1) by 3. 3 (x + y) = 3 5 => 3x + 3y = 15 (Equation 3) Step 2: Add Equation (3) and Equation (2). (3x + 3y) + (2x - 3y) = 15 + 4 5x = 19 x = 19/5 Step 3: Substitute x = 19/5 into Equation (1). 19/5 + y = 5 y = 5 - 19/5 y = (25 - 19)/5 y = 6/5 Step 4: Verify the solution (optional, but good practice). For Eq (1): 19/5 + 6/5 = 25/5 = 5 (True) For Eq (2): 2(19/5) - 3(6/5) = 38/5 - 18/5 = 20/5 = 4 (True) Final Answer: x = 19/5, y = 6/5
  • Example 2: Solve the following pair of linear equations by the elimination method: Equation (1): 3x + 4y = 10 Equation (2): 2x - 2y = 2 Solution: Step 1: We want to eliminate 'y'. The coefficients are 4 and -2. We can make them 4 and -4 (or 4 and 4) by multiplying Equation (2) by 2. 2 (2x - 2y) = 2 2 => 4x - 4y = 4 (Equation 3) Step 2: Add Equation (1) and Equation (3). (3x + 4y) + (4x - 4y) = 10 + 4 7x = 14 x = 14/7 x = 2 Step 3: Substitute x = 2 into Equation (1). 3(2) + 4y = 10 6 + 4y = 10 4y = 10 - 6 4y = 4 y = 1 Step 4: Verify the solution. For Eq (1): 3(2) + 4(1) = 6 + 4 = 10 (True) For Eq (2): 2(2) - 2(1) = 4 - 2 = 2 (True) Final Answer: x = 2, y = 1

Exam Tip: Avoiding Common Mistakes in Elimination

When using the Elimination Method, students often make a few common errors. Be very careful with signs: if you're subtracting equations, remember to change the sign of every term in the equation being subtracted. A frequent mistake is to only change the sign of the first term. Also, when multiplying an equation by a constant to make coefficients equal, ensure you multiply all terms on both sides of the equation, not just the term you're trying to adjust. Lastly, always perform a quick check by substituting your final x and y values back into both original equations. This simple step can save you from losing marks on silly calculation errors!

Practice Questions with Solutions

  • Q: Solve the following pair of linear equations by the elimination method: x + 2y = 7 2x - y = 4 A: Step 1: To eliminate 'y', multiply the second equation by 2: 2 (2x - y) = 2 4 => 4x - 2y = 8 (Eq 3) Step 2: Add Eq (1) and Eq (3): (x + 2y) + (4x - 2y) = 7 + 8 5x = 15 x = 3 Step 3: Substitute x = 3 into Eq (1): 3 + 2y = 7 2y = 4 y = 2 Final answer: x = 3, y = 2
  • Q: Solve using the elimination method: 3x - 5y - 4 = 0 9x = 2y + 7 A: Step 1: Rearrange both equations into the standard form ax + by = c: Eq (1): 3x - 5y = 4 Eq (2): 9x - 2y = 7 Step 2: To eliminate 'x', multiply Eq (1) by 3: 3 (3x - 5y) = 3 4 => 9x - 15y = 12 (Eq 3) Step 3: Subtract Eq (3) from Eq (2): (9x - 2y) - (9x - 15y) = 7 - 12 9x - 2y - 9x + 15y = -5 13y = -5 y = -5/13 Step 4: Substitute y = -5/13 into Eq (1): 3x - 5(-5/13) = 4 3x + 25/13 = 4 3x = 4 - 25/13 3x = (52 - 25)/13 3x = 27/13 x = 9/13 Final answer: x = 9/13, y = -5/13
  • Q: Two numbers differ by 3. The sum of twice the smaller number and the larger number is 19. Find the numbers. A: Step 1: Let the larger number be 'x' and the smaller number be 'y'. From the first condition: x - y = 3 (Eq 1) From the second condition: 2y + x = 19 (Eq 2) Step 2: We can eliminate 'x' by subtracting Eq (1) from Eq (2): (x + 2y) - (x - y) = 19 - 3 x + 2y - x + y = 16 3y = 16 y = 16/3 Step 3: Substitute y = 16/3 into Eq (1): x - 16/3 = 3 x = 3 + 16/3 x = (9 + 16)/3 x = 25/3 Final answer: The larger number is 25/3 and the smaller number is 16/3.
  • Q: Solve for x and y: (x/2) + (2y/3) = -1 x - (y/3) = 3 A: Step 1: Simplify the equations to remove fractions: For Eq (1): Multiply by LCM(2,3)=6: 6 (x/2) + 6 (2y/3) = 6 (-1) => 3x + 4y = -6 (Eq 3) For Eq (2): Multiply by LCM(1,3)=3: 3 (x) - 3 (y/3) = 3 3 => 3x - y = 9 (Eq 4) Step 2: To eliminate 'x', subtract Eq (4) from Eq (3): (3x + 4y) - (3x - y) = -6 - 9 3x + 4y - 3x + y = -15 5y = -15 y = -3 Step 3: Substitute y = -3 into Eq (4): 3x - (-3) = 9 3x + 3 = 9 3x = 6 x = 2 Final answer: x = 2, y = -3

Frequently Asked Questions

What is the main idea behind the Elimination Method?

The main idea is to eliminate one of the two variables by making their coefficients numerically equal and then either adding or subtracting the equations. This leaves you with a single equation in one variable, which is much easier to solve.

When should I use addition and when subtraction in the Elimination Method?

You should add the equations if the equal coefficients of the variable you're eliminating have opposite signs (e.g., +2y and -2y). You should subtract the equations if the equal coefficients have the same sign (e.g., +3x and +3x, or -5y and -5y).

Can I always eliminate 'x' first, or must I eliminate 'y'?

You can choose to eliminate either 'x' or 'y' first. The choice often depends on which variable's coefficients are easier to make equal by multiplication, requiring smaller or simpler factors. The final solution will be the same regardless of which variable you eliminate first.

What if, after elimination, I get 0 = 0 or a false statement like 0 = 5?

If you get 0 = 0 after elimination, it means the system has infinitely many solutions (the lines are coincident). If you get a false statement like 0 = 5, it means the system has no solution (the lines are parallel and distinct). These are special cases you might encounter.