We are given the system of equations:

1. $y = \frac{2}{3}x + 3$
2. $y = -x - 7$

Step 1: Find the Intersection (Solution)

Since we are solving by graphing, we look for the point where these two lines intersect.

Graphing Each Line

1. First Equation: $y = \frac{2}{3}x + 3$

   • Slope: $\frac{2}{3}$ (rise 2, run 3)
   • y-intercept: $(0,3)$

2. Second Equation: $y = -x - 7$

   • Slope: $-1$ (rise -1, run 1)
   • y-intercept: $(0,-7)$

Step 2: Solve Algebraically (To Confirm)

Set the two equations equal to each other: $\frac{2}{3}x + 3 = -x - 7$

Multiply everything by 3 to eliminate the fraction: $2x + 9 = -3x - 21$

Solve for $x$: $2x + 3x = -21 - 9$ $5x = -30$ $x = -6$

Substituting $x = -6$ into one of the original equations: $y = -(-6) - 7$ $y = 6 - 7 = -1$

Final Answer:

The solution to the system is $(-6,-1)$.

Would you like a visual graph of the solution? Let me know if you need details!

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Further Questions:

1. How do you determine the slope from an equation?
2. How do you find the point of intersection without graphing?
3. What happens if two lines have the same slope but different y-intercepts?
4. How would the solution change if one equation was quadratic?
5. What does it mean if a system has no solution?

Tip: When graphing, always check the y-intercept first, then use the slope to find additional points!