NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Example 1

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Example 1

Question:

Check graphically whether the pair of equations \( x + 3y = 6 \) and \( 2x - 3y = 12 \) are consistent.

Given:

The pair of linear equations:
1. \( x + 3y = 6 \)
2. \( 2x - 3y = 12 \)

To Find:

To check graphically whether the given pair of equations is consistent (i.e., if they have a solution or not).

Formula:

  • To solve graphically, plot both equations on the Cartesian plane.
  • If the lines intersect at a point, the system is consistent and has a unique solution.
  • If the lines are parallel, the system is inconsistent (no solution).
  • If the lines coincide, there are infinitely many solutions.

Solution:

Step 1: Write the equations in standard form.

$$ \begin{align*} \text{Equation 1:} & \quad x + 3y = 6 \\ \text{Equation 2:} & \quad 2x - 3y = 12 \end{align*} $$

Step 2: Find at least two solutions for each equation.

For Equation 1: \( x + 3y = 6 \)

  • If \( x = 0 \): \( 0 + 3y = 6 \implies y = \frac{6}{3} = 2 \)
  • If \( y = 0 \): \( x + 0 = 6 \implies x = 6 \)

So, points: \( (0, 2) \) and \( (6, 0) \).

For Equation 2: \( 2x - 3y = 12 \)

  • If \( x = 0 \): \( 0 - 3y = 12 \implies -3y = 12 \implies y = \frac{12}{-3} = -4 \)
  • If \( y = 0 \): \( 2x - 0 = 12 \implies 2x = 12 \implies x = \frac{12}{2} = 6 \)

So, points: \( (0, -4) \) and \( (6, 0) \).

Step 3: Prepare a table of values for both equations.

$$ \begin{array}{|c|c|c|} \hline \text{Equation} & x & y \\ \hline x + 3y = 6 & 0 & 2 \\ x + 3y = 6 & 6 & 0 \\ 2x - 3y = 12 & 0 & -4 \\ 2x - 3y = 12 & 6 & 0 \\ \hline \end{array} $$

Step 4: Plot the points and draw the lines on the graph.

  • For \( x + 3y = 6 \), plot points \( (0, 2) \) and \( (6, 0) \).
  • For \( 2x - 3y = 12 \), plot points \( (0, -4) \) and \( (6, 0) \).

Both lines pass through \( (6, 0) \).

Step 5: Observe the graph.

The two lines intersect at the point \( (6, 0) \).

Therefore, the pair of equations is consistent and has a unique solution at \( (6, 0) \).

Result:

The given pair of equations is consistent and has a unique solution:
\( x = 6, \; y = 0 \).
(The lines intersect at the point \( (6, 0) \) on the graph.)

© Kaliyuga Ekalavya. All rights reserved.

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