Question

This graph shows the lines for these equations:
3y=3x + 6
y = x + 2
How many solutions are there for this system of equations?

No solutions

One solution

Infinite solutions

Answer 1

To determine how many solutions there are for the system of equations, we first need to rewrite both equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

1. The first equation is \(3y = 3x + 6\). We can divide every term by 3 to simplify it:
\[
y = x + 2
\]

2. The second equation is already in slope-intercept form:
\[
y = x + 2
\]

Now, we see that both equations are identical:
\[
y = x + 2
\]
This means they represent the same line.

Since both equations yield the same line, they coincide, leading to an infinite number of solutions. Every point on the line \(y = x + 2\) is a solution to the system.

Thus, the answer is **Infinite solutions**.