To solve the problem, we are asked to find the coordinates of point $P$, where line $L$ is parallel to the line given by $y = 4x + 3$.

Key Information:

• The equation of the given line is $y = 4x + 3$. This is in slope-intercept form $y = mx + b$, where the slope $m = 4$ and the y-intercept $b = 3$.
• Line $L$ is parallel to this line, which means it will have the same slope, $m = 4$, but a different y-intercept.
• From the graph, line $L$ passes through the point $(0, -3)$ (as marked in the image). This means we can use this point to find the equation of line $L$.

Step 1: Find the Equation of Line $L$

Since line $L$ has the same slope as the line $y = 4x + 3$, its equation will also be in the form $y = 4x + b$. We can use the point $(0, -3)$, which lies on line $L$, to find the value of $b$.

Substitute $x = 0$ and $y = -3$ into the equation: $-3 = 4(0) + b$ $b = -3$ So, the equation of line $L$ is: $y = 4x - 3$

Step 2: Find the Coordinates of Point $P$

Point $P$ is the intersection of line $L$ and the vertical line $x = 1$ (as shown in the graph). To find the y-coordinate of point $P$, substitute $x = 1$ into the equation of line $L$.

$y = 4(1) - 3$ $y = 4 - 3 = 1$

Thus, the coordinates of point $P$ are $(1, 1)$.

Final Answer:

The coordinates of point $P$ are $\mathbf{(1, 1)}$.

Would you like more details or have any further questions on this? Here are some related questions you could explore:

1. What happens to the equation of a line when it becomes perpendicular instead of parallel?
2. How do you determine if two lines are parallel by their equations alone?
3. Can you find the distance between two parallel lines?
4. How would the equation of line $L$ change if it passed through a different point?
5. What is the general form of the equation of a line?

Tip: When lines are parallel, they always have the same slope but different y-intercepts unless they are identical!