Question

On a coordinate plane, the x-axis is labeled width of photo and the y-axis is labeled length of photo. Line a goes through points (1, 2) and (2, 4). Line b goes through points (4, 5) and (6, 7.5). Line C goes through points (5, 4) and (10, 8). Lind d goes through points (2, 1) and (4, 2).

line a
line b
line c
line d

Answer 1

To analyze the lines given the points, we can determine their equations and slopes.

1. **Line a**:
- Points: (1, 2) and (2, 4)
- Slope \(m\) = \(\frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 2}{2 - 1} = \frac{2}{1} = 2\)
- Using point-slope form (y - y₁ = m(x - x₁)), we can use (1, 2):
\[
y - 2 = 2(x - 1) \\
y - 2 = 2x - 2 \\
y = 2x
\]

2. **Line b**:
- Points: (4, 5) and (6, 7.5)
- Slope \(m\) = \(\frac{7.5 - 5}{6 - 4} = \frac{2.5}{2} = 1.25\)
- Using point (4, 5) to find the equation:
\[
y - 5 = 1.25(x - 4) \\
y - 5 = 1.25x - 5 \\
y = 1.25x
\]

3. **Line c**:
- Points: (5, 4) and (10, 8)
- Slope \(m\) = \(\frac{8 - 4}{10 - 5} = \frac{4}{5}\)
- Using point (5, 4):
\[
y - 4 = \frac{4}{5}(x - 5) \\
y - 4 = \frac{4}{5}x - 4 \\
y = \frac{4}{5}x
\]

4. **Line d**:
- Points: (2, 1) and (4, 2)
- Slope \(m\) = \(\frac{2 - 1}{4 - 2} = \frac{1}{2}\)
- Using point (2, 1):
\[
y - 1 = \frac{1}{2}(x - 2) \\
y - 1 = \frac{1}{2}x - 1 \\
y = \frac{1}{2}x
\]

### Summary of the equations:
- Line a: \(y = 2x\)
- Line b: \(y = 1.25x\)
- Line c: \(y = \frac{4}{5}x\)
- Line d: \(y = \frac{1}{2}x\)

### Interpretation:
- Line a has the steepest slope (2), followed by line b (1.25), then line c (0.8), and finally line d (0.5) which is the least steep.

If you need further analysis or visualization, please let me know!