Solve Graphically the System of Linear Equations and Find the Area Bounded by the Lines and the Y-Axis: 3x + y − 11 = 0, x − y − 1 = 0

Video Explanation

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

Solution

Question:
Solve the following system of linear equations graphically and shade the region bounded by these lines and the y-axis. Also, find the area of the region bounded by these lines and the y-axis:
3x + y − 11 = 0
x − y − 1 = 0

Step 1: Rewrite the Equations in Slope-Intercept Form

For 3x + y − 11 = 0:

y = 11 − 3x

For x − y − 1 = 0:

−y = −x + 1
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 3x + y − 11 = 0:

Putting x = 0:
y = 11
So, the line meets the y-axis at (0, 11).

For x − y − 1 = 0:

Putting x = 0:
−y − 1 = 0 ⇒ 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:

11 − 3x = x − 1

4x = 12 ⇒ x = 3

Substituting x = 3 in y = x − 1:

y = 2

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

Step 4: Vertices of the Region and Shading

The region is bounded by the two given lines and the y-axis.

The vertices of the bounded region are:

(0, 11), (0, −1) and (3, 2).

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

Step 5: Calculate the Area of the Region

The base of the triangle lies on the y-axis between y = −1 and y = 11.

Length of base = 11 − (−1) = 12 units

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

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

Area = (1/2) × 12 × 3 = 18 square units

Final Answer

∴ The graphical solution of the given system of equations is (3, 2).

The area bounded by the given lines and the y-axis is 18 square units.

Conclusion

Thus, the two straight lines and the y-axis form a triangular region with vertices (0, 11), (0, −1) and (3, 2), and the area of the region is 18 square units.