Solve Graphically the System of Linear Equations and Shade the Region Between the Lines and the X-Axis: 3x + 2y − 4 = 0, 2x − 3y − 7 = 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 between the two lines and the x-axis:
3x + 2y − 4 = 0
2x − 3y − 7 = 0

Step 1: Rewrite the Equations in Slope-Intercept Form

For 3x + 2y − 4 = 0:

2y = 4 − 3x
y = 2 − (3/2)x

For 2x − 3y − 7 = 0:

−3y = −2x + 7
y = (2/3)x − 7/3

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

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

For 3x + 2y − 4 = 0:

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

For 2x − 3y − 7 = 0:

Putting y = 0:
2x − 7 = 0 ⇒ x = 7/2
So, the line meets the x-axis at (7/2, 0).

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

Solving the equations simultaneously:

3x + 2y = 4
2x − 3y = 7

Multiplying first equation by 3 and second by 2:

9x + 6y = 12
4x − 6y = 14

Adding both equations:

13x = 26 ⇒ x = 2

Substituting x = 2 in 3x + 2y = 4:

6 + 2y = 4 ⇒ 2y = −2 ⇒ y = −1

So, the point of intersection is (2, −1).

Step 4: Graphical Interpretation and Shading of Region

When the graphs of the given equations are drawn on the same Cartesian plane, the two straight lines intersect at the point (2, −1).

The x-axis, the line 3x + 2y − 4 = 0, and the line 2x − 3y − 7 = 0 together enclose a triangular region.

The region bounded by these two lines and the x-axis is shaded.

Final Answer

∴ The graphical solution of the given system of equations is (2, −1).

The shaded region is the triangular region enclosed by the lines 3x + 2y − 4 = 0, 2x − 3y − 7 = 0, and the x-axis.

Conclusion

Since the two straight lines intersect at one point, the system of linear equations has a unique solution. The region between the two lines and the x-axis forms a triangle which is shaded.