Question

Solve the system of linear equations: $x = 4$ and $3x - 2y = 6$ graphically.

Hint:

To verify a graphical solution, always substitute the intersection coordinates back into both equations.
For $(4, 3)$:
- $x = 4$ (True)
- $3(4) - 2(3) = 12 - 6 = 6$ (True)
This guarantees your graphical plot is perfectly accurate!

Answer 1

Step 1: Find the intersection point algebraically first, as a check for the graph.
Substitute $x=4$ directly into the second equation: \[ 3(4) - 2y = 6 \] \[ 12 - 2y = 6 \] \[ 2y = 6 \] \[ y = 3 \] So the two lines must meet at $(4,3)$.
Step 2: Plot the line x = 4.
This is a vertical line through every point with x-coordinate $4$, such as $(4,0)$ and $(4,-3)$.
Step 3: Plot the line 3x - 2y = 6 using two points.
At $x=0$, $y=-3$, giving $(0,-3)$; at $x=2$, $y=0$, giving $(2,0)$. Draw the straight line through these points.
Step 4: Read off the intersection.
The two lines cross exactly at the point found algebraically in Step 1, confirming the graphical solution.
\[ \boxed{(x,y) = (4,3)} \]