The graph in the image displays two linear equations:

• The red line represents $y = x$.
• The blue line represents $x + 2y = 6$.

To find the solution, we determine the point of intersection of these two lines. Here's how we solve it algebraically:

Step 1: Solve the blue equation for $y$

From $x + 2y = 6$, subtract $x$:
$2y = 6 - x$ Divide by 2: $y = 3 - \frac{x}{2}$

Step 2: Substitute $y = x$ into $y = 3 - \frac{x}{2}$

Replace $y$ with $x$ in the second equation:
$x = 3 - \frac{x}{2}$ Multiply through by 2 to eliminate the fraction:
$2x = 6 - x$ Add $x$ to both sides:
$3x = 6$ Divide by 3:
$x = 2$

Step 3: Solve for $y$

Since $y = x$, and $x = 2$:
$y = 2$

Final Answer:

The solution to the system is the point $(2, 2)$. This is where the two lines intersect.
Correct choice: $(2, 2)$.

Would you like further clarification on any steps?

Relative Questions:

1. How can you graphically verify the solution to a system of linear equations?
2. What happens when two lines in a system are parallel?
3. How do you determine the number of solutions a system of linear equations has?
4. Can substitution always be used to solve a system of equations? Why or why not?
5. How does the slope of each line relate to the solution of a system?

Tip:

When solving a system of equations graphically, always double-check the coordinates of the intersection point by plugging them back into the original equations.