Question

Graph the equations y=2x+1 and y=-3/2x+8

Answer 1

To graph the equations \( y = 2x + 1 \) and \( y = -\frac{3}{2}x + 8 \), we can follow these steps:

### Step 1: Graph the line \( y = 2x + 1 \)

1. **Identify the y-intercept**: The y-intercept (where the line crosses the y-axis) is \( b = 1 \). So, the point is \( (0, 1) \).

2. **Identify the slope**: The slope \( m = 2 \) means that for every 1 unit increase in \( x \), \( y \) increases by 2 units.

3. **Plot another point using the slope**: Starting from \( (0, 1) \):
- When \( x = 1 \), \( y = 2(1) + 1 = 3 \). Plot the point \( (1, 3) \).
- When \( x = -1 \), \( y = 2(-1) + 1 = -1 \). Plot the point \( (-1, -1) \).

4. **Draw the line**: Connect the points \( (0, 1) \) and \( (1, 3) \) to draw the line.

### Step 2: Graph the line \( y = -\frac{3}{2}x + 8 \)

1. **Identify the y-intercept**: The y-intercept is \( b = 8 \). So, the point is \( (0, 8) \).

2. **Identify the slope**: The slope \( m = -\frac{3}{2} \) means that for every 2 units increase in \( x \), \( y \) decreases by 3 units.

3. **Plot another point using the slope**: Starting from \( (0, 8) \):
- When \( x = 2 \), \( y = -\frac{3}{2}(2) + 8 = 5 \). Plot the point \( (2, 5) \).
- When \( x = -2 \), \( y = -\frac{3}{2}(-2) + 8 = 11 \). Plot the point \( (-2, 11) \).

4. **Draw the line**: Connect the points \( (0, 8) \) and \( (2, 5) \) to draw the line.

### Step 3: Find the Intersection

To find where the two lines intersect, set the equations equal to each other:

\[
2x + 1 = -\frac{3}{2}x + 8
\]

Multiply everything by 2 to eliminate the fraction:

\[
4x + 2 = -3x + 16
\]

Combine like terms:

\[
4x + 3x = 16 - 2
\]

\[
7x = 14 \implies x = 2
\]

Now substitute \( x = 2 \) into one of the original equations to find \( y \):

\[
y = 2(2) + 1 = 5
\]

So, the intersection point is \( (2, 5) \).

### Final Graph Summary

- Line \( y = 2x + 1 \) crosses at \( (0, 1) \) and \( (1, 3) \).
- Line \( y = -\frac{3}{2}x + 8 \) crosses at \( (0, 8) \) and \( (2, 5) \).
- Both lines intersect at the point \( (2, 5) \).

You can now sketch the graph using these points and information! Would you like a drawn out graph or further assistance?