Question

Consider the following set of equations:

Answer 1

Part 1:

For this case we have the following equations:

$-3y = -3x - 9\\y = x + 3$

We can rewrite the system of equations:

To do this, we divide equation 1 between -3:

$\frac{-3}{-3}y = \frac{-3}{-3}x - \frac{9}{-3}$

Rewriting we have:

$y = x + 3$

Therefore, we observe that we have two equal linear equations.

Thus, the lines are cut into infinite points.

The system then has endless solutions:

Answer:

Infinite solutions

Part 2:

For this case we have the following equations:

$y = x + 4\\y = 3x + 6$

We observe that we have two lines with different slopes, therefore, the system has only one solution.

The solution is obtained by matching both equations:

$x + 4 = 3x + 6$

Answer:

The following is a step that can be used to find the solution to the set of equations:

$x + 4 = 3x + 6$

Answer 2

1.
$-3y=-3x-9 \\ y=x+3 \\ \\ \hbox{substitute x+3 for y in the first equation:} \\ -3(x+3)=-3x-9 \\ -3x-9=-3x-9 \\ -3x+3x=-9+9 \\ 0=0 \\ always \ true \\ \\ \boxed{\hbox{infinitely many solutions}} \Rightarrow "infinite \ solutions"$

2.
$y=x+4 \\ y=3x+6 \\ \\ y=y \\ \boxed{x+4=3x+6} \Leftarrow \hbox{the step} \\ x-3x=6-4 \\ -2x=2 \\ x=\frac{2}{-2} \\ x=-1 \\ \\ y=x+4 \\ y=-1+4 \\ y=3 \\ \\ (x,y)=(-1,3)$