Question

Graph the system of equations below
y=-1/2x-1
y=1/4x-4

Answer 1

To graph the system of equations given by:

1. \( y = -\frac{1}{2}x - 1 \)
2. \( y = \frac{1}{4}x - 4 \)

### Step 1: Graph the first equation

**Equation 1:** \( y = -\frac{1}{2}x - 1 \)

- **Slope**: \(-\frac{1}{2}\)
- **Y-intercept**: \(-1\) (the point (0, -1))

Find another point using the slope:
- Start at (0, -1). Since the slope is -1/2, for every 2 units you move right (positive x), move 1 unit down (negative y).
- From (0, -1) to (2, -1.5).

Plot the points (0, -1) and (2, -1.5) and draw a line through them.

### Step 2: Graph the second equation

**Equation 2:** \( y = \frac{1}{4}x - 4 \)

- **Slope**: \(\frac{1}{4}\)
- **Y-intercept**: \(-4\) (the point (0, -4))

Find another point using the slope:
- Start at (0, -4). Since the slope is 1/4, for every 4 units you move right (positive x), move 1 unit up (positive y).
- From (0, -4) to (4, -3).

Plot the points (0, -4) and (4, -3) and draw a line through them.

### Step 3: Finding the intersection

To find the intersection point (if it exists), set the equations equal to each other to solve for \(x\):

\[
-\frac{1}{2}x - 1 = \frac{1}{4}x - 4
\]

Multiplying through by 4 to eliminate fractions:

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

Combine the \(x\) terms and constant terms:

\[
-2x - x = -16 + 4
\]
\[
-3x = -12 \implies x = 4
\]

Now substitute \(x = 4\) back into either equation to find \(y\):

Using the second equation:

\[
y = \frac{1}{4}(4) - 4 = 1 - 4 = -3
\]

Thus, the intersection point is \((4, -3)\).

### Final Graph

- The first line has a negative slope and crosses the y-axis at (0, -1).
- The second line has a positive slope and crosses the y-axis at (0, -4).
- They intersect at the point (4, -3).

You can plot both lines and mark the intersection point to visualize the solution of the system of equations! If you have graphing paper or software, it can help you accurately represent the lines.