https://file.aiquickdraw.com/mathbot/file/1727645714627nveu9p67.jpg

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.

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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 \(-1\).

1. La ecuación de una recta perpendicular a la recta 2x + 5y = 13. 2. La ecuación de una recta paralela a la recta -4x + y = 3.

Learn how to find the equation of a line perpendicular or parallel to a given line. Solve the linear equations using slope-intercept form and apply the rules of perpendicularity and parallelism. Suitable for Grades 9-12.

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