Draw the Graphs and Find the Vertices and Area of the Triangle Formed by x − y + 1 = 0, 3x + 2y − 12 = 0 and the X-Axis

Video Explanation

Watch the video below to understand the complete solution step by step:

Solution

Question:
Draw the graphs of the following equations:
x − y + 1 = 0
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. Also calculate the area of the triangle.

Step 1: Rewrite the Equations in Slope-Intercept Form

For x − y + 1 = 0:

−y = −x − 1
y = x + 1

For 3x + 2y − 12 = 0:

2y = 12 − 3x
y = 6 − (3/2)x

Step 2: Find the Points Where the Lines Meet the X-Axis

A line meets the x-axis where y = 0.

For x − y + 1 = 0:

Putting y = 0:
x + 1 = 0 ⇒ x = −1
So, the line meets the x-axis at (−1, 0).

For 3x + 2y − 12 = 0:

Putting y = 0:
3x − 12 = 0 ⇒ x = 4
So, the line meets the x-axis at (4, 0).

Step 3: Find the Point of Intersection of the Two Lines

Solving the equations simultaneously:

y = x + 1
y = 6 − (3/2)x

Equating the two values of y:

x + 1 = 6 − (3/2)x
(5/2)x = 5 ⇒ x = 2

Substituting x = 2 in y = x + 1:

y = 3

So, the point of intersection is (2, 3).

Step 4: Vertices of the Triangle and Shading

The triangle is formed by the two given lines and the x-axis.

The vertices of the triangle are:

(−1, 0), (4, 0) and (2, 3).

The triangular region enclosed by these lines and the x-axis is shaded.

Step 5: Calculate the Area of the Triangle

The base of the triangle lies on the x-axis between x = −1 and x = 4.

Length of base = 4 − (−1) = 5 units

Height of the triangle = y-coordinate of the vertex opposite the base = 3 units

Area of triangle = (1/2) × base × height

Area = (1/2) × 5 × 3 = 15/2 square units

Final Answer

∴ The vertices of the triangle are (−1, 0), (4, 0) and (2, 3).

The area of the triangular region bounded by the given lines and the x-axis is 15/2 square units.

Conclusion

Thus, the given two straight lines and the x-axis form a triangle whose vertices are (−1, 0), (4, 0) and (2, 3), and the area of the triangle is 15/2 square units.