To solve this, we will determine the equation of line $B$ that is parallel to line $A$ and passes through point $P$. Let’s break it down step by step:

Step 1: Identify the slope ($m$) of line $A$

• From the graph, we can observe that line $A$ goes through two points: $(-8, -5)$ and $(0, -1)$.
• Using the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the points: $m = \frac{-1 - (-5)}{0 - (-8)} = \frac{-1 + 5}{8} = \frac{4}{8} = \frac{1}{2}$

Step 2: Write the equation of line $B$

Since line $B$ is parallel to line $A$, it has the same slope $m = \frac{1}{2}$. Line $B$ passes through point $P = (-3, 2)$. Using the point-slope form of a line: $y - y_1 = m(x - x_1)$ Substituting $m = \frac{1}{2}$, $x_1 = -3$, and $y_1 = 2$: $y - 2 = \frac{1}{2}(x - (-3))$ $y - 2 = \frac{1}{2}(x + 3)$ Simplify: $y - 2 = \frac{1}{2}x + \frac{3}{2}$ $y = \frac{1}{2}x + \frac{3}{2} + 2$ Convert $2$ to a fraction with a denominator of 2: $y = \frac{1}{2}x + \frac{3}{2} + \frac{4}{2}$ $y = \frac{1}{2}x + \frac{7}{2}$

Final Answer:

The equation of line $B$ is: $y = \frac{1}{2}x + \frac{7}{2}$

Would you like me to clarify any of the steps or go into further details?

Here are some related questions:

1. How do you find the equation of a perpendicular line?
2. How do you convert between different forms of linear equations?
3. What is the significance of the slope in real-world scenarios?
4. How do you find the intersection of two lines given their equations?
5. How do you graph a line using its slope and $y$-intercept?

Tip: When writing equations, always simplify fractions to their lowest terms for clarity and accuracy.