Solve the equations using the graphical method:

x + y = 3
-3x + 2y = 1
Options:
(A) (1, -2)
(B) (-1, 2)
(C) (1, 2)
(D) (-1, -2)

To solve the given system of linear equations using the graphical method, we will first rewrite both equations in the slope-intercept form (y = mx + b) to plot them on a graph.

The given equations are:

x + y = 3
-3x + 2y = 1
Step 1: Rewrite the equations in slope-intercept form.
Equation 1: x + y = 3

To rewrite this in slope-intercept form, solve for y: y = 3 - x

Now, the equation is y = -x + 3.

Equation 2: -3x + 2y = 1

Solve for y: 2y = 3x + 1 y = (3/2)x + 1/2

Now, the equation is y = (3/2)x + 1/2.

Step 2: Plot the lines on a graph.
For the first equation, y = -x + 3, the slope is -1 and the y-intercept is 3. So, the line crosses the y-axis at (0, 3), and the slope tells us that for every 1 unit increase in x, y decreases by 1.

For the second equation, y = (3/2)x + 1/2, the slope is 3/2 and the y-intercept is 1/2. So, the line crosses the y-axis at (0, 1/2), and for every 2 units increase in x, y increases by 3 units.

Step 3: Find the point of intersection.
To find the point of intersection, we can solve the two equations algebraically.

From the first equation, y = -x + 3. Substitute this into the second equation: -3x + 2(-x + 3) = 1 -3x - 2x + 6 = 1 -5x + 6 = 1 -5x = 1 - 6 -5x = -5 x = 1

Substitute x = 1 into the first equation: x + y = 3 1 + y = 3 y = 3 - 1 y = 2

Step 4: Conclusion
The point of intersection is (1, 2). Therefore, the solution to the system of equations is (1, 2).

Answer: (C). (1, 2)