Question

Graph the system of linear equations and then state how many solutions.
y=3x+5
y=3x+2

Answer 1

To graph the system of equations given, we have:

1. \( y = 3x + 5 \)
2. \( y = 3x + 2 \)

### Step 1: Graph the first equation
For the equation \( y = 3x + 5 \):
- The y-intercept is \( 5 \), so the point \( (0, 5) \) is on the graph.
- The slope is \( 3 \), which means for every 1 unit increase in \( x \), \( y \) increases by \( 3 \). So another point can be \( (1, 8) \) (since \( y = 3(1) + 5 = 8 \)).

### Step 2: Graph the second equation
For the equation \( y = 3x + 2 \):
- The y-intercept is \( 2 \), so the point \( (0, 2) \) is on the graph.
- The slope is also \( 3 \), so another point can be \( (1, 5) \) (since \( y = 3(1) + 2 = 5 \)).

### Step 3: Analyzing the lines
Both equations have the same slope of \( 3 \) but different y-intercepts. This means the lines are parallel to each other.

### Step 4: Conclusion
Since the lines are parallel and will never intersect, the system of equations has **no solutions**.

So, the final statement is:

**The system of equations has no solutions.**