NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 7

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

Question:

Draw the graphs of the equations \(x - y + 1 = 0\) and \(3x + 2y - 12 = 0\). Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.

Given:

  • Equation 1: \(x - y + 1 = 0\)
  • Equation 2: \(3x + 2y - 12 = 0\)
  • The x-axis (\(y = 0\))

To Find:

  • The coordinates of the vertices of the triangle formed by the two lines and the x-axis.
  • Graphical representation with the triangular region shaded.

Formula:

  • To find intersection of two lines: Solve the two equations simultaneously.
  • To find intersection with x-axis: Substitute \(y = 0\) in the equation of the line.

Solution:

Step 1: Find the x-intercept of \(x - y + 1 = 0\).

Substitute \(y = 0\):
\[ x - 0 + 1 = 0 \implies x + 1 = 0 \implies x = -1 \] So, the intersection with the x-axis is \((-1, 0)\).

Step 2: Find the x-intercept of \(3x + 2y - 12 = 0\).

Substitute \(y = 0\):
\[ 3x + 2 \times 0 - 12 = 0 \implies 3x - 12 = 0 \implies 3x = 12 \implies x = 4 \] So, the intersection with the x-axis is \((4, 0)\).

Step 3: Find the intersection point of the two lines.

Solve the equations: \[ \begin{align*} x - y + 1 &= 0 \quad \cdots (1) \\ 3x + 2y - 12 &= 0 \quad \cdots (2) \end{align*} \] From (1): \(x = y - 1\)
Substitute in (2): \[ 3(y - 1) + 2y - 12 = 0 \\ 3y - 3 + 2y - 12 = 0 \\ 5y - 15 = 0 \implies y = 3 \] Now, substitute \(y = 3\) in (1): \[ x - 3 + 1 = 0 \implies x - 2 = 0 \implies x = 2 \] So, the intersection point is \((2, 3)\).

Step 4: List the vertices of the triangle.

The triangle is formed by the points:
  • \(A(-1, 0)\): Intersection of \(x - y + 1 = 0\) with x-axis
  • \(B(4, 0)\): Intersection of \(3x + 2y - 12 = 0\) with x-axis
  • \(C(2, 3)\): Intersection of the two lines

Step 5: Draw the graph and shade the triangular region.

Graph of the triangle formed by x – y + 1 = 0, 3x + 2y – 12 = 0, and the x-axis
The triangle is shaded and its vertices are \((-1, 0)\), \((4, 0)\), and \((2, 3)\).

Result:

The coordinates of the vertices of the triangle formed by the lines \(x - y + 1 = 0\), \(3x + 2y - 12 = 0\), and the x-axis are:
  • \((-1, 0)\)
  • \((4, 0)\)
  • \((2, 3)\)
The triangle formed by these points is shown shaded in the graph above.
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