Solve Graphically the System of Linear Equations and Find the Area Bounded by the Lines and the X-Axis: 4x − 3y + 4 = 0, 4x + 3y − 20 = 0

Video Explanation

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

Solution

Question:
Solve graphically the following system of linear equations and find the area bounded by these lines and the x-axis:
4x − 3y + 4 = 0
4x + 3y − 20 = 0

Step 1: Rewrite the Equations in Slope-Intercept Form

For 4x − 3y + 4 = 0:

−3y = −4x − 4
y = (4x + 4)/3

For 4x + 3y − 20 = 0:

3y = 20 − 4x
y = (20 − 4x)/3

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

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

For 4x − 3y + 4 = 0:

Putting y = 0:
4x + 4 = 0 ⇒ x = −1
So, the line meets the x-axis at (−1, 0).

For 4x + 3y − 20 = 0:

Putting y = 0:
4x − 20 = 0 ⇒ x = 5
So, the line meets the x-axis at (5, 0).

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

Solving the equations simultaneously:

y = (4x + 4)/3
y = (20 − 4x)/3

Equating the two values of y:

4x + 4 = 20 − 4x
8x = 16 ⇒ x = 2

Substituting x = 2 in y = (4x + 4)/3:

y = 4

So, the point of intersection is (2, 4).

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), (5, 0) and (2, 4).

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 = 5.

Length of base = 5 − (−1) = 6 units

Height of the triangle = y-coordinate of the vertex opposite the base = 4 units

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

Area = (1/2) × 6 × 4 = 12 square units

Final Answer

∴ The graphical solution of the given system of equations is (2, 4).

The area bounded by the given lines and the x-axis is 12 square units.

Conclusion

Thus, the two straight lines and the x-axis form a triangle whose vertices are (−1, 0), (5, 0) and (2, 4), and the area of the triangle is 12 square units.