For each pair of linear equations given below, drawn graphs and then state, whether the lines drawn are parallel or perpendicular to each other.
`y=x-3`
`y=-x+5`

Answer

To solve the problem of determining whether the lines represented by the equations \( y = x - 3 \) and \( y = -x + 5 \) are parallel or perpendicular, we can follow these steps:

### Step 1: Identify the slopes of the equations
The equations are in slope-intercept form \( y = mx + b \), where \( m \) is the slope.

1. For the first equation \( y = x - 3 \):
- The slope \( m_1 = 1 \).

2. For the second equation \( y = -x + 5 \):
- The slope \( m_2 = -1 \).

### Step 2: Determine the relationship between the slopes
To check if the lines are parallel or perpendicular:
- Lines are **parallel** if their slopes are equal (\( m_1 = m_2 \)).
- Lines are **perpendicular** if the product of their slopes is \(-1\) (\( m_1 \cdot m_2 = -1 \)).

Calculating the product of the slopes:
\[ m_1 \cdot m_2 = 1 \cdot (-1) = -1 \]

### Step 3: State the conclusion
Since the product of the slopes is \(-1\), the lines are perpendicular to each other.

### Step 4: Graph the equations
To visualize the lines, we can plot them on a graph.

1. **Graph of \( y = x - 3 \)**:
- When \( x = 0 \), \( y = -3 \) (point: \( (0, -3) \)).
- When \( y = 0 \), \( x = 3 \) (point: \( (3, 0) \)).
- Plot these points and draw the line.

2. **Graph of \( y = -x + 5 \)**:
- When \( x = 0 \), \( y = 5 \) (point: \( (0, 5) \)).
- When \( y = 0 \), \( x = 5 \) (point: \( (5, 0) \)).
- Plot these points and draw the line.

### Final Conclusion
After graphing both equations, we can visually confirm that the lines intersect at a right angle, confirming they are perpendicular.

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