Let's analyze the two questions in the image.

Problem 1:

It asks for the equation of a line perpendicular to the given line $2x + 5y = 13$.

1. First, rewrite the given equation $2x + 5y = 13$ in slope-intercept form $y = mx + b$, where $m$ is the slope: $5y = -2x + 13 \quad \Rightarrow \quad y = \frac{-2}{5}x + \frac{13}{5}$ The slope of this line is $\frac{-2}{5}$.

2. The slope of a line perpendicular to this will be the negative reciprocal of $\frac{-2}{5}$, which is $\frac{5}{2}$.

3. Now, we check which of the given options has a slope of $\frac{5}{2}$:

   • Option C has the equation $y = \frac{5}{2}x + 4$, which has the correct slope.

Thus, Option C is the correct answer.

Problem 2:

It asks for the equation of a line parallel to the given line $-4x + y = 3$.

1. Again, rewrite the given equation $-4x + y = 3$ in slope-intercept form $y = mx + b$: $y = 4x + 3$ The slope of this line is 4.

2. For parallel lines, the slopes must be the same. So, we are looking for an equation with a slope of 4.

3. Let's check the options:

   • Option A has the equation $y = 4x - 5$, which has the correct slope of 4.

Thus, Option A is the correct answer.

Would you like further details on any part of this solution?

Here are 5 related questions to practice:

1. How do you find the slope of a line given in standard form?
2. What is the relationship between the slopes of parallel lines?
3. How do you find the negative reciprocal of a fraction?
4. How can you rewrite an equation from standard form to slope-intercept form?
5. How do you determine if two lines are perpendicular?

Tip: When two lines are perpendicular, their slopes multiply to $-1$.