To solve the system of equations by graphing, we need to plot both equations on the same coordinate plane and find their point of intersection.

The system of equations is:

1. $x + y = 2$
2. $x - 2y = 8$

Step 1: Rewrite Each Equation in Slope-Intercept Form

Equation 1: $x + y = 2$ Solve for $y$: $y = -x + 2$

Equation 2: $x - 2y = 8$ Solve for $y$: $-2y = -x + 8 \implies y = \frac{1}{2}x - 4$

Step 2: Graph Both Equations

1. Graph the first equation $y = -x + 2$:

   • The y-intercept is 2 (point: $(0, 2)$).
   • The slope is -1, so from the y-intercept, move down 1 unit and right 1 unit to plot the next point (point: $(1, 1)$).

2. Graph the second equation $y = \frac{1}{2}x - 4$:

   • The y-intercept is -4 (point: $(0, -4)$).
   • The slope is $\frac{1}{2}$, so from the y-intercept, move up 1 unit and right 2 units to plot the next point (point: $(2, -3)$).

Step 3: Find the Point of Intersection

Now we need to find the point where these two lines intersect. Let's solve the system algebraically to find the intersection point:

Set the equations equal to each other: $-x + 2 = \frac{1}{2}x - 4$

Combine like terms: $-2x + 4 = x - 8$

Simplify and solve for $x$: $-2x - x = -8 - 4 \implies -3x = -12 \implies x = 4$

Substitute $x = 4$ back into one of the equations (e.g., $y = -x + 2$): $y = -4 + 2 = -2$

Solution:

The point of intersection is $(4, -2)$.

Therefore, the solution to the system of equations is: $\boxed{(4, -2)}$

Would you like me to provide a graph to visualize this solution? Also, feel free to ask any of the following questions:

1. How do you graph linear equations with different slopes?
2. Can you explain more about the slope-intercept form of a line?
3. How do you determine the solution of a system of equations by graphing?
4. What happens if two lines are parallel when solving by graphing?
5. How can I check if my solution is correct after solving graphically?

Tip: Always double-check your plotted points to ensure accuracy when solving systems by graphing!