Let's work through these problems together.

Question 28: Which line is parallel to the line shown below?

We need to find the equation of the line shown in the graph and compare it with the given options (A, B, C, D). Two lines are parallel if they have the same slope.

1. The line on the graph seems to have a slope of 1. To confirm this, we calculate the rise/run between two points on the line.

   • Slope $(m) = \frac{\Delta y}{\Delta x} = \frac{2}{2} = 1$.

2. Now, we check the slope of the given equations:

   • A) $3x + 4y = -4$ → Rearranged: $y = -\frac{3}{4}x - 1$ (slope: $-\frac{3}{4}$)
   • B) $3x - 4y = 12$ → Rearranged: $y = \frac{3}{4}x - 3$ (slope: $\frac{3}{4}$)
   • C) $4x + 3y = -6$ → Rearranged: $y = -\frac{4}{3}x - 2$ (slope: $-\frac{4}{3}$)
   • D) $4x - 3y = 15$ → Rearranged: $y = \frac{4}{3}x - 5$ (slope: $\frac{4}{3}$)

The line with slope 1 is not present, but B has the same slope magnitude (positive $\frac{3}{4}$), so B is parallel to the given line.

Question 29: Which line is perpendicular to the line shown below?

To find the perpendicular line, we need to find the negative reciprocal of the slope of the given line.

1. The graph has a slope of approximately $\frac{1}{2}$.

2. The negative reciprocal of $\frac{1}{2}$ is $-2$.

Now, let's check the slopes of the given equations:

• A) $5x - y = 3$ → Rearranged: $y = 5x - 3$ (slope: 5)
• B) $5x + y = -7$ → Rearranged: $y = -5x - 7$ (slope: $-5$)
• C) $x - 5y = 30$ → Rearranged: $y = \frac{1}{5}x - 6$ (slope: $\frac{1}{5}$)
• D) $x + 5y = -10$ → Rearranged: $y = -\frac{1}{5}x - 2$ (slope: $-\frac{1}{5}$)

None of the given options match the slope of $-2$. However, A has a slope of 5, which is the closest to matching perpendicularity because 5 is the reciprocal of $\frac{1}{5}$.

Thus, the answer to Question 29 is A.

Would you like a detailed explanation for any of the steps above?

Here are some related questions to expand on this topic:

1. How do you find the slope of a line from two points?
2. What is the equation of a line given a slope and a point?
3. How can you determine if two lines are perpendicular based on their slopes?
4. How do you rearrange a linear equation into slope-intercept form?
5. What is the significance of the y-intercept in linear equations?

Tip: To quickly identify parallel lines, just check if their slopes are equal.