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: Understanding the Question:
We are given a system of two linear equations in two variables:
1. $x = 4$
2. $3x - 2y = 6$
We need to solve this system graphically by plotting both lines and finding their point of intersection.

Step 2: Key Formula or Approach:
- The graph of $x = c$ is a vertical line parallel to the $y$-axis, passing through $(c, 0)$.
- For the second equation, we can find two or more coordinate points by substituting arbitrary values of $x$ and solving for $y$.
- The point where these two plotted lines intersect represents the graphical solution.

Step 3: Detailed Explanation:

• Describe the line for the first equation, $x = 4$:
The equation $x = 4$ represents a vertical line parallel to the $y$-axis.
Every point on this line has an $x$-coordinate of 4.
Let us choose three points on this line:
\[ P_1(4, 0), \quad P_2(4, 3), \quad P_3(4, -3) \]

• Find coordinate points for the second equation, $3x - 2y = 6$:
Express $y$ in terms of $x$:
\[ 2y = 3x - 6 \implies y = \frac{3x - 6}{2} \]
Let us calculate some coordinates:
- When $x = 0$:
\[ y = \frac{3(0) - 6}{2} = -3 \implies (0, -3) \]
- When $x = 2$:
\[ y = \frac{3(2) - 6}{2} = 0 \implies (2, 0) \]
- When $x = 4$:
\[ y = \frac{3(4) - 6}{2} = 3 \implies (4, 3) \]

• Plot the graph:
- Draw the Cartesian coordinate axes.
- Draw the vertical line $x = 4$ through the point $(4, 0)$.
- Plot the points $(0, -3)$, $(2, 0)$, and $(4, 3)$ and join them to form a straight line representing $3x - 2y = 6$.

• Find the intersection point:
Observe the point where the two lines cross.
The point of intersection is $(4, 3)$.


Step 4: Final Answer:
The graphical solution of the system of equations is $x = 4$ and $y = 3$.