Let's analyze each of the graphs to determine if lines $\ell_1$ and $\ell_2$ are parallel.

1. Determining if lines are parallel

To check if two lines are parallel, we can compare their slopes. If two lines have the same slope, they are parallel.

The slope $m$ of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Now, let's compute the slopes for each pair of lines in the exercises:

Exercise 7:

• Line $\ell_1$:

  • Points: $(-6, 1)$ and $(0, -2)$
  • Slope: $m_1 = \frac{-2 - 1}{0 - (-6)} = \frac{-3}{6} = -\frac{1}{2}$

• Line $\ell_2$:

  • Points: $(0, 3)$ and $(4, 1)$
  • Slope: $m_2 = \frac{1 - 3}{4 - 0} = \frac{-2}{4} = -\frac{1}{2}$

Since both slopes are $-\frac{1}{2}$, lines $\ell_1$ and $\ell_2$ are parallel.

Exercise 8:

• Line $\ell_1$:

  • Points: $(-5, 1)$ and $(-4, -2)$
  • Slope: $m_1 = \frac{-2 - 1}{-4 - (-5)} = \frac{-3}{1} = -3$

• Line $\ell_2$:

  • Points: $(1, 3)$ and $(4, 2)$
  • Slope: $m_2 = \frac{2 - 3}{4 - 1} = \frac{-1}{3} = -\frac{1}{3}$

Since the slopes are $-3$ and $-\frac{1}{3}$, lines $\ell_1$ and $\ell_2$ are not parallel.

Exercise 9:

• Line $\ell_1$:

  • Points: $(-4, -4)$ and $(0, 2)$
  • Slope: $m_1 = \frac{2 - (-4)}{0 - (-4)} = \frac{6}{4} = \frac{3}{2}$

• Line $\ell_2$:

  • Points: $(2, -3)$ and $(5, 3)$
  • Slope: $m_2 = \frac{3 - (-3)}{5 - 2} = \frac{6}{3} = 2$

Since the slopes are $\frac{3}{2}$ and $2$, lines $\ell_1$ and $\ell_2$ are not parallel.

Exercise 10:

• Line $\ell_1$:

  • Points: $(-3, 1)$ and $(-2, 5)$
  • Slope: $m_1 = \frac{5 - 1}{-2 - (-3)} = \frac{4}{1} = 4$

• Line $\ell_2$:

  • Points: $(0, -2)$ and $(1, 2)$
  • Slope: $m_2 = \frac{2 - (-2)}{1 - 0} = \frac{4}{1} = 4$

Since both slopes are $4$, lines $\ell_1$ and $\ell_2$ are parallel.

Summary of Parallelism:

• Exercise 7: Parallel
• Exercise 8: Not parallel
• Exercise 9: Not parallel
• Exercise 10: Parallel

Would you like any more details or have further questions?

Follow-up Questions:

1. How do you find the slope of a vertical line?
2. What happens if two lines have slopes that are negative reciprocals of each other?
3. Can two lines that are not parallel ever have the same slope?
4. How can you determine if two lines are perpendicular using their slopes?
5. What happens if the slope of one line is zero? What does this represent?

Tip: When comparing slopes, ensure to simplify fractions, as parallel lines must have identical slopes in their simplest form!