To solve this problem, we need to determine the equation of a line that is parallel to Line $A$ and passes through point $P$.

Step 1: Determine the Slope of Line $A$

1. Identify two points on Line $A$ from the graph. Let's choose:

   • Point $(0, -2)$
   • Point $(2, 1)$

2. Calculate the slope $m$ of Line $A$ using the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the values: $m = \frac{1 - (-2)}{2 - 0} = \frac{3}{2}$

So, the slope of Line $A$ is $\frac{3}{2}$.

Step 2: Determine the Equation of the Line Passing Through Point $P$ and Parallel to Line $A$

Since the line we want is parallel to Line $A$, it will have the same slope, $m = \frac{3}{2}$.

We are given that this line passes through point $P$ at $(1, 3)$.

Step 3: Use the Point-Slope Form to Find the Equation

The point-slope form of a line is: $y - y_1 = m(x - x_1)$ Substituting $m = \frac{3}{2}$, $x_1 = 1$, and $y_1 = 3$: $y - 3 = \frac{3}{2}(x - 1)$

Step 4: Simplify the Equation

Expand and simplify to put it in the form $y = mx + c$: $y - 3 = \frac{3}{2}x - \frac{3}{2}$ $y = \frac{3}{2}x - \frac{3}{2} + 3$ $y = \frac{3}{2}x + \frac{3}{2}$

Final Answer

The equation of the line parallel to Line $A$ that passes through point $P$ is: $y = \frac{3}{2}x + \frac{3}{2}$

Would you like further details on any part of this solution? Here are some related questions to explore:

1. How can you find the equation of a line perpendicular to Line $A$ passing through point $P$?
2. What is the y-intercept of the new line compared to Line $A$?
3. How would the equation change if point $P$ were different?
4. How can you graphically verify the parallelism of two lines?
5. How do you determine if two lines are parallel from their equations alone?

Tip: Remember that parallel lines always have the same slope, so finding the slope is crucial in these types of problems.