Question

A system of equations is shown.
y = 2.5x - 5
y = -0.75x + 3

I need to graph the equations

Answer 1

To graph the system of equations given as:

1. \( y = 2.5x - 5 \)
2. \( y = -0.75x + 3 \)

First, we can identify the slope and y-intercept for each equation to plot them.

### Equation 1: \( y = 2.5x - 5 \)
- **Slope (m)**: 2.5
- **Y-intercept (b)**: -5 (this is the point (0, -5))

#### Points for Graphing:
- When \( x = 0 \): \( y = -5 \) → point (0, -5)
- When \( x = 2 \): \( y = 2.5(2) - 5 = 0 \) → point (2, 0)
- When \( x = -2 \): \( y = 2.5(-2) - 5 = -10 \) → point (-2, -10)

### Equation 2: \( y = -0.75x + 3 \)
- **Slope (m)**: -0.75
- **Y-intercept (b)**: 3 (this is the point (0, 3))

#### Points for Graphing:
- When \( x = 0 \): \( y = 3 \) → point (0, 3)
- When \( x = 4 \): \( y = -0.75(4) + 3 = 0 \) → point (4, 0)
- When \( x = -4 \): \( y = -0.75(-4) + 3 = 6 \) → point (-4, 6)

### Steps to Graph the Equations:

1. **Plot the points** for both equations on a coordinate plane.
2. **Draw a line** through the points for the first equation (from \( (0, -5) \) to \( (2, 0) \)).
3. **Draw a line** through the points for the second equation (from \( (0, 3) \) to \( (4, 0) \)).
4. **Find the intersection point**, if applicable, which is the solution to the system of equations.

#### Intersection Point:
To find where the two lines intersect, we can set the equations equal to each other:

\[
2.5x - 5 = -0.75x + 3
\]

Combining like terms:

\[
2.5x + 0.75x = 3 + 5
\]
\[
3.25x = 8
\]
\[
x = \frac{8}{3} \approx 2.67
\]

To find \( y \), substitute \( x \) back into either equation. Using the first equation:

\[
y = 2.5(2.67) - 5
\]
\[
y \approx 6.675 - 5 \approx 1.675
\]

So the intersection point is approximately \( (2.67, 1.68) \).

Now you can graph the two lines and note the intersection point!