Draw the Graphs of 5x − y = 5 and 3x − y = 3 and Find the Area of the Triangle Formed

Video Explanation

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

Solution

Question: Draw the graphs of the equations 5x − y = 5 and 3x − y = 3. Determine the coordinates of the vertices of the triangle formed by these lines and the y-axis. Also, calculate the area of the triangle so formed.

Step 1: Write the Equations in y-form

Given equations:

5x − y = 5 ⇒ y = 5x − 5

3x − y = 3 ⇒ y = 3x − 3

Step 2: Find Points on Each Line

For the line y = 5x − 5:

When x = 0, y = −5 ⇒ Point A (0, −5)

When y = 0, x = 1 ⇒ Point B (1, 0)

For the line y = 3x − 3:

When x = 0, y = −3 ⇒ Point C (0, −3)

When y = 0, x = 1 ⇒ Point D (1, 0)

Step 3: Plot the Graphs

Plot the points A (0, −5) and B (1, 0) to draw the line y = 5x − 5.

Plot the points C (0, −3) and D (1, 0) to draw the line y = 3x − 3.

Both lines intersect the y-axis and meet at the point (1, 0).

Step 4: Coordinates of the Triangle Vertices

The triangle is formed by the two lines and the y-axis. The vertices are:

A (0, −5), C (0, −3), B (1, 0)

Step 5: Calculate the Area of the Triangle

Base = Distance between A and C on y-axis = |−3 − (−5)| = 2 units

Height = Distance of point B from y-axis = 1 unit

Area of triangle = ½ × base × height = ½ × 2 × 1 = 1 square unit

Final Answer

∴ The coordinates of the vertices of the triangle are (0, −5), (0, −3), and (1, 0).

∴ The area of the triangle formed is 1 square unit.

Conclusion

Thus, by plotting the graphs of the given linear equations and identifying their intercepts with the y-axis, we find that the area of the triangle formed is 1 square unit.