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

Video Explanation

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

Solution

Question:
Draw the graphs of the following equations on the same graph paper:
2x + 3y = 12
x − y = 1
Find the coordinates of the vertices of the triangle formed by the two straight lines and the y-axis.

Step 1: Rewrite the Equations in Slope-Intercept Form

For 2x + 3y = 12:

3y = 12 − 2x
y = 4 − (2/3)x

For x − y = 1:

−y = 1 − x
y = x − 1

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

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

For 2x + 3y = 12:

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

For x − y = 1:

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

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

Solving the equations simultaneously:

2x + 3y = 12
x − y = 1

From x − y = 1 ⇒ y = x − 1

Substituting in 2x + 3y = 12:

2x + 3(x − 1) = 12
5x − 3 = 12 ⇒ 5x = 15 ⇒ x = 3

Substituting x = 3 in y = x − 1:

y = 2

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

Step 4: Graphical Interpretation

When the graphs of the two given equations are drawn on the same Cartesian plane, they intersect at the point (3, 2).

The y-axis intersects the two lines at the points (0, 4) and (0, −1).

These three points form a triangle.

Final Answer

∴ The vertices of the triangle formed by the two straight lines and the y-axis are:
(0, 4), (0, −1) and (3, 2).

Conclusion

Thus, the given two straight lines and the y-axis form a triangle whose vertices are (0, 4), (0, −1) and (3, 2).