Solve Graphically the System of Linear Equations and Shade the Region Between the Lines and the X-Axis: 2x + 3y = 12, x − y = 1

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:
2x + 3y = 12
x − y = 1

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 X-Axis

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

For 2x + 3y = 12:

Putting y = 0:
2x = 12 ⇒ x = 6
So, the line meets the x-axis at (6, 0).

For x − y = 1:

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

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 is (3, 2).

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 (3, 2).

The x-axis, the line 2x + 3y = 12, and the line x − y = 1 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 (3, 2).

The shaded region is the triangular region enclosed by the lines 2x + 3y = 12, x − y = 1, 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.