Determine by Drawing Graphs Whether the System of Linear Equations Has a Unique Solution: 2x − 3y = 6, x + y = 1

Video Explanation

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

Solution

Question:
Determine, by drawing graphs, whether the following system of linear equations has a unique solution or not:
2x − 3y = 6
x + y = 1

Step 1: Rewrite the Equations in Slope-Intercept Form

For 2x − 3y = 6:

−3y = 6 − 2x
y = (2/3)x − 2

For x + y = 1:

y = 1 − x

Step 2: Compare the Two Equations

Slope of first line = 2/3
Slope of second line = −1

Since the slopes of the two lines are different, the lines are not parallel.

Step 3: Graphical Interpretation

When the graphs of both equations are drawn on the same Cartesian plane, the two straight lines intersect at exactly one point.

Hence, the given system of linear equations has a unique solution.

Step 4: Find the Point of Intersection

Solving the two equations, we get:

x = 3 and y = −2

Final Answer

∴ The given system of linear equations has a unique solution.

Conclusion

Since the two lines intersect at one point, the system of linear equations is consistent and has a unique solution.