Solve the system of equations y = 3x + 3 y = 10x + 10 (-1,0) (-1,-8) (4,0) (0, -1)

1. Break Down the Problem

We have a system of two equations:

1. $y = 3x + 3$
2. $y = 10x + 10$

We need to find the point(s) where these two lines intersect, which will be our solution.

2. Relevant Concepts

To find the intersection point, we can set the two equations equal to each other since they both equal $y$: $3x + 3 = 10x + 10$

3. Analysis and Detail

Let's solve for $x$:

1. Rearranging the equation, we have: $3x + 3 - 10x - 10 = 0$ This simplifies to: $-7x - 7 = 0$
2. Adding $7$ to both sides: $-7x = 7$
3. Dividing by $-7$: $x = -1$

Now we can substitute $x = -1$ back into either equation to find $y$. Let's use the first equation: $y = 3(-1) + 3 = -3 + 3 = 0$

So, the intersection point is: $(-1, 0)$

4. Verify and Summarize

Let's verify by substituting $(-1, 0)$ into the second equation: $y = 10(-1) + 10 = -10 + 10 = 0$ Since it satisfies both equations, our solution is correct.

Final Answer

The solution to the system of equations is the point $(-1, 0)$.