NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 1(ii)

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

Question:

Form the pair of linear equations in the following problems, and find their solutions graphically.
5 pencils and 7 pens together cost ₹50, whereas 7 pencils and 5 pens together cost ₹46. Find the cost of one pencil and that of one pen.

Given:

  • 5 pencils and 7 pens together cost ₹50.
  • 7 pencils and 5 pens together cost ₹46.

To Find:

  • The cost of one pencil.
  • The cost of one pen.

Formula:

  • Let the cost of one pencil be ₹\( x \).
  • Let the cost of one pen be ₹\( y \).
  • Linear equation in two variables: \( ax + by = c \).
  • Graphical solution: The intersection point of the two lines represents the solution.

Solution:

Step 1: Let the cost of one pencil be ₹\( x \) and the cost of one pen be ₹\( y \).

According to the question:

  • 5 pencils and 7 pens: \( 5x + 7y = 50 \)
  • 7 pencils and 5 pens: \( 7x + 5y = 46 \)

So, the pair of linear equations is:

$$ \begin{align*} 5x + 7y &= 50 \quad \cdots (1) \\ 7x + 5y &= 46 \quad \cdots (2) \end{align*} $$

Step 2: Find two points for each equation to plot their graphs.

For equation (1): \( 5x + 7y = 50 \)

  • If \( x = 0 \): \( 7y = 50 \implies y = \frac{50}{7} \approx 7.14 \)
  • If \( y = 0 \): \( 5x = 50 \implies x = \frac{50}{5} = 10 \)
  • If \( x = 3 \): \( 5 \times 3 + 7y = 50 \implies 15 + 7y = 50 \implies 7y = 35 \implies y = 5 \)

Points for (1): \( (0, 7.14),\ (10, 0),\ (3, 5) \)

Step 3: Find two points for the second equation.

For equation (2): \( 7x + 5y = 46 \)

  • If \( x = 0 \): \( 5y = 46 \implies y = \frac{46}{5} = 9.2 \)
  • If \( y = 0 \): \( 7x = 46 \implies x = \frac{46}{7} \approx 6.57 \)
  • If \( x = 4 \): \( 7 \times 4 + 5y = 46 \implies 28 + 5y = 46 \implies 5y = 18 \implies y = \frac{18}{5} = 3.6 \)

Points for (2): \( (0, 9.2),\ (6.57, 0),\ (4, 3.6) \)

Step 4: Plot the points for both equations on a graph and draw the lines.

The intersection point of the two lines gives the solution to the pair of equations.

Step 5: Solve the equations algebraically to find the exact values.

Multiply equation (1) by 7 and equation (2) by 5:

$$ \begin{align*} 7 \times (5x + 7y) &= 7 \times 50 \implies 35x + 49y = 350 \quad (3) \\ 5 \times (7x + 5y) &= 5 \times 46 \implies 35x + 25y = 230 \quad (4) \end{align*} $$

Subtract (4) from (3):

$$ (35x + 49y) - (35x + 25y) = 350 - 230 \\ 24y = 120 \implies y = \frac{120}{24} = 5 $$

Step 6: Substitute \( y = 5 \) in equation (1) to find \( x \).

$$ 5x + 7 \times 5 = 50 \\ 5x + 35 = 50 \\ 5x = 15 \implies x = \frac{15}{5} = 3 $$

Step 7: Therefore, the solution is:

  • Cost of one pencil = ₹3
  • Cost of one pen = ₹5

On the graph, the two lines intersect at the point \( (3, 5) \).

Result:

  • The cost of one pencil is ₹3.
  • The cost of one pen is ₹5.
  • The solution is the point \( (3, 5) \) on the graph.
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