To solve this system of equations by graphing, let's analyze the given equations:

1. $y = -\frac{2}{3}x - 5$

   • This equation represents a line with a slope of $-\frac{2}{3}$ and a y-intercept of $-5$.

2. $y = -1$

   • This equation represents a horizontal line where $y$ is constantly $-1$, regardless of the $x$-value.

Solution Steps:

1. Graph the First Equation $y = -\frac{2}{3}x - 5$:

   • Start at the y-intercept, which is $-5$ (point $(0, -5)$).
   • Use the slope $-\frac{2}{3}$ to find another point on the line. The slope tells us that for every 3 units we move to the right, we move 2 units down.
   • Moving from $(0, -5)$:
     • Right 3 units: $x = 3$
     • Down 2 units: $y = -7$
     • So another point is $(3, -7)$.

2. Graph the Second Equation $y = -1$:

   • This line is horizontal and crosses the y-axis at $y = -1$.
   • The line passes through all points with $y = -1$, such as $(0, -1)$, $(1, -1)$, and $(-1, -1)$.

3. Find the Intersection Point:

   • The intersection of these two lines represents the solution to the system of equations.
   • Substitute $y = -1$ from the second equation into the first equation to find $x$: $-1 = -\frac{2}{3}x - 5$
   • Solve for $x$: $-1 + 5 = -\frac{2}{3}x$ $4 = -\frac{2}{3}x$ $x = -6$
   • So, the point of intersection is $(-6, -1)$.

Solution:

The solution to the system of equations is: $(x, y) = (-6, -1)$

Would you like a detailed graph for further clarity or have any other questions?

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Follow-up Questions:

1. How can we verify the solution of a system of equations by substitution?
2. What are some other methods to solve systems of equations besides graphing?
3. How does changing the slope in an equation affect the graph of the line?
4. What is the significance of parallel lines in a system of equations?
5. Can systems of equations have no solutions or infinitely many solutions? Under what conditions?

Tip:

When solving systems by graphing, ensure that both lines are drawn accurately, as even a small miscalculation can affect where they intersect.