Question

Identify the number of solutions to the system of equation:
y = ⅓ x + 2
-x + 3y = 6

How can you tell from the equations?

Finish graphing:

Answer 1

To identify the number of solutions to the system of equations:

1. **Equation 1:** \( y = \frac{1}{3}x + 2 \)
This is in slope-intercept form (y = mx + b), where the slope \( m \) is \( \frac{1}{3} \) and the y-intercept \( b \) is 2.

2. **Equation 2:** \( -x + 3y = 6 \)
We can rewrite this in slope-intercept form by solving for \( y \):

\[
-x + 3y = 6 \\
3y = x + 6 \\
y = \frac{1}{3}x + 2
\]

Now, we can see that both equations are equivalent:

- From Equation 1: \( y = \frac{1}{3}x + 2 \)
- From Equation 2: \( y = \frac{1}{3}x + 2 \)

Since both equations represent the same line, there are **infinitely many solutions** to this system of equations, as every point on the line is a solution.

### Conclusion

You can tell there are infinitely many solutions because both equations represent the same line in the coordinate plane. Therefore, they intersect at every point along that line.

### Graphing

To graph the equations:

1. The first equation \( y = \frac{1}{3}x + 2 \) has a y-intercept of \( 2 \) (the point \( (0, 2) \)) and a slope of \( \frac{1}{3} \) (rise over run). For every 3 units you move right (positive x-direction), you move up 1 unit (positive y-direction).

2. The second equation, after re-arranging, gives you the same line. Thus, it will overlap exactly with the first line.

You can plot the following points based on the equation \( y = \frac{1}{3}x + 2 \):

- When \( x = 0 \), \( y = 2 \) (point \( (0, 2) \)).
- When \( x = 3 \), \( y = \frac{1}{3}(3) + 2 = 3 \) (point \( (3, 3) \)).
- When \( x = -3 \), \( y = \frac{1}{3}(-3) + 2 = 1 \) (point \( (-3, 1) \)).

Plotting these points along with drawing a straight line through them will represent both equations, which overlap perfectly. The graph will show a single line, confirming that there are infinitely many solutions.