Question

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


How many solutions does the system of linear equations have? ___

Answer 1

To graph the system of linear equations, we will plot the two lines represented by the equations:

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

Both equations are in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

### Step 1: Finding the slopes and intercepts

- For the first equation \(y = 3x + 5\):
- Slope (\(m\)) = 3
- Y-intercept (\(b\)) = 5 (the point where the line crosses the y-axis is (0,5))

- For the second equation \(y = 3x + 2\):
- Slope (\(m\)) = 3
- Y-intercept (\(b\)) = 2 (the point where the line crosses the y-axis is (0,2))

### Step 2: Plotting the lines

1. For \(y = 3x + 5\):
- At \(x = 0\), \(y = 5\) → point (0, 5)
- At \(x = 1\), \(y = 3(1) + 5 = 8\) → point (1, 8)

2. For \(y = 3x + 2\):
- At \(x = 0\), \(y = 2\) → point (0, 2)
- At \(x = 1\), \(y = 3(1) + 2 = 5\) → point (1, 5)

### Step 3: Analyzing the lines

- Both lines have the same slope (3), which means they are parallel.
- Because they have different y-intercepts (5 and 2), they will never intersect.

### Conclusion

Since the two lines are parallel and will never meet, the system of linear equations has **no solutions**.

Answer: **0 solutions**