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

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. 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.

Given:

  • Total number of students = 10
  • The number of girls is 4 more than the number of boys

To Find:

  • The number of boys
  • The number of girls
  • Form the pair of linear equations and solve them graphically

Formula:

  • Let the number of boys = \( x \)
  • Let the number of girls = \( y \)
  • According to the question:
    • \( x + y = 10 \)
    • \( y = x + 4 \)
  • To solve graphically: Plot both equations on the coordinate plane. The intersection point gives the solution.

Solution:

Step 1: Let the number of boys be \( x \) and the number of girls be \( y \).

$$ x = \text{number of boys} $$ $$ y = \text{number of girls} $$

Step 2: Form the linear equations using the given information.

  • Total number of students: \( x + y = 10 \)   ...(1)
  • Number of girls is 4 more than number of boys: \( y = x + 4 \)   ...(2)

Step 3: Rewrite equation (2) in standard form.

$$ y = x + 4 $$ $$ y - x = 4 \qquad ...(2') $$

Step 4: Find two points for each equation to plot them on the graph.

For \( x + y = 10 \):
If \( x = 2 \), \( y = 8 \)
If \( x = 6 \), \( y = 4 \)
Points: \( (2, 8) \), \( (6, 4) \)

For \( y - x = 4 \):
If \( x = 2 \), \( y = 6 \)
If \( x = 4 \), \( y = 8 \)
Points: \( (2, 6) \), \( (4, 8) \)

Step 5: Solve the equations algebraically to find the intersection point.

$$ \begin{align*} x + y &= 10 \quad ...(1) \\ y - x &= 4 \quad ...(2') \end{align*} $$ Add equations (1) and (2'): $$ (x + y) + (y - x) = 10 + 4 $$ $$ 2y = 14 $$ $$ y = \frac{14}{2} = 7 $$

Step 6: Substitute \( y = 7 \) in equation (1) to find \( x \).

$$ x + 7 = 10 $$ $$ x = 10 - 7 = 3 $$

Step 7: Verification of the solution.

  • Number of boys = \( 3 \)
  • Number of girls = \( 7 \)
  • Is the number of girls 4 more than boys? \( 7 - 3 = 4 \) ✔️
  • Total students: \( 3 + 7 = 10 \) ✔️

Step 8: Graphical representation.

When you plot the lines \( x + y = 10 \) and \( y - x = 4 \) on the coordinate plane, they intersect at the point \( (3, 7) \).

Therefore, the number of boys is 3 and the number of girls is 7.
(You can plot the graph for practice.)

Result:

  • Number of boys who took part in the quiz = 3
  • Number of girls who took part in the quiz = 7
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