Question

determine if lines r and s are parallel.
select the correct choice below and fill in the answer box(es) to complete your choice.
a. the slope of line r is \\(\square\\), and the slope of line s is \\(\square\\). since the product of their slopes is \\(-1\\), lines r and s are parallel.
(simplify your answers.)
b. since both lines have a slope of \\(\square\\), lines r and s are parallel.
(simplify your answer.)
c. the slope of line r is \\(\square\\), and the slope of line s is \\(\square\\). since the product of their slopes is not \\(-1\\), lines r and s are not parallel.
(simplify your answers.)
d. the slope of line r is \\(\square\\), and the slope of line s is \\(\square\\). since the slopes are different, lines r and s are not parallel.
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Explanation:

To determine if lines \( r \) and \( s \) are parallel, we first need to find their slopes using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Step 1: Find two points on line \( r \)

From the graph, let's identify two points on line \( r \). Suppose we pick \( (x_1, y_1) = (0, 8) \) and \( (x_2, y_2) = (4, 0) \).

Step 2: Calculate the slope of line \( r \)

Using the slope formula:

$$ m_r = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 8}{4 - 0} = \frac{-8}{4} = -2 $$

Step 3: Find two points on line \( s \)

From the graph, let's identify two points on line \( s \). Suppose we pick \( (x_1, y_1) = (0, 6) \) and \( (x_2, y_2) = (3, 0) \).

Step 4: Calculate the slope of line \( s \)

Using the slope formula:

$$ m_s = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 6}{3 - 0} = \frac{-6}{3} = -2 $$

Step 5: Compare the slopes

Parallel lines have equal slopes. Here, \( m_r = -2 \) and \( m_s = -2 \), so the slopes are equal.

Lines are parallel if their slopes are equal. We calculated the slope of line \( r \) as \( -2 \) and the slope of line \( s \) as \( -2 \). Since the slopes are equal, lines \( r \) and \( s \) are parallel.

Answer:

B. Since both lines have a slope of \( \boldsymbol{-2} \), lines \( r \) and \( s \) are parallel.