To solve the system of equations by graphing, we find the point where the two lines intersect. Let’s begin with the given equations:

1. $y = x$
2. $y = -2x + 9$

Steps:

1. Graph the First Equation $y = x$:

   • This is a line with a slope of 1, passing through the origin $(0, 0)$.
   • A couple of points to plot: $(0, 0)$, $(1, 1)$, $(-1, -1)$.

2. Graph the Second Equation $y = -2x + 9$:

   • This is a line with a slope of $-2$ and a y-intercept of $9$.
   • A couple of points to plot: $(0, 9)$, $(1, 7)$, $(2, 5)$.

3. Find the Intersection Point:

   • The intersection is the solution of the system. To find it algebraically, set the two equations equal: $x = -2x + 9$ Solve for $x$: $x + 2x = 9 \implies 3x = 9 \implies x = 3$
   • Substitute $x = 3$ into $y = x$: $y = 3$
   • So, the solution is $(3, 3)$.

4. Check the Solution:

   • Substitute $(3, 3)$ into both equations:
     • For $y = x$: $3 = 3$ ✅
     • For $y = -2x + 9$: $3 = -2(3) + 9 \implies 3 = 3$ ✅

Thus, the solution is verified.

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Final Answer:

The solution to the system is: $\boxed{(3, 3)}$

Would you like a graph to visualize this? Let me know!

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Related Questions:

1. What does the slope of a line represent in a linear equation?
2. How do you find the intersection point of two lines algebraically?
3. What are the possible outcomes for a system of linear equations?
4. How does graphing help solve a system of equations compared to substitution or elimination?
5. Can a system of equations have no solution or infinitely many solutions? How?

Tip:

When graphing lines, always calculate at least two points to ensure accuracy and confirm the line's direction.