Let's go through each question and solve them one by one:

Problem #1

The system of equations is: $y = -\frac{2}{3}x - 1$ $4x + 6y = -6$

1. Substitute $y = -\frac{2}{3}x - 1$ into the second equation: $4x + 6\left(-\frac{2}{3}x - 1\right) = -6$

2. Distribute the $6$ in the second equation: $4x - 4x - 6 = -6$

3. After simplification, this becomes $-6 = -6$, which is always true. Therefore, this system has infinitely many solutions.

Answer for #1: Option 4) Infinitely many solutions

Problem #2

The system of equations is: $x + 3y = 10$ $x = 2y - 5$

1. Substitute $x = 2y - 5$ into the first equation: $(2y - 5) + 3y = 10$

2. Simplify and solve for $y$: $5y - 5 = 10$ $5y = 15$ $y = 3$

3. Substitute $y = 3$ back into $x = 2y - 5$: $x = 2(3) - 5$ $x = 6 - 5 = 1$

So, the solution is $(1, 3)$.

Answer for #2: Option 1) (1, 3)

Problem #3

From the graph, we need to find the intersection point of the two lines.

Observing the graph, the lines intersect at the point $(2, -3)$.

Answer for #3: Option 3) (2, -3)

If you’d like further explanation on any of these solutions, let me know!

Related Questions:

1. How can you determine if a system of equations has no solution, one solution, or infinitely many solutions?
2. What are other methods for solving systems of linear equations besides substitution and graphing?
3. Why do some systems result in statements like $-6 = -6$, and what does that imply about the solution?
4. How can you check if a point is a solution to a system of equations without graphing?
5. What are the differences between consistent, inconsistent, and dependent systems?

Tip: When dealing with equations, always simplify as much as possible to make substitution or elimination easier.