Let's go through each question step-by-step.

Question 1

Find the slope using the points (5, 3) and (7, 2):

The formula for the slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Using $(x_1, y_1) = (5, 3)$ and $(x_2, y_2) = (7, 2)$: $m = \frac{2 - 3}{7 - 5} = \frac{-1}{2} = -\frac{1}{2}$

So, the slope between the points (5, 3) and (7, 2) is $-\frac{1}{2}$.

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Question 2

Find the slopes and determine if the lines are parallel, perpendicular, or intersecting:

Step 1: Find the slope between $(6, -9)$ and $(1, -8)$:

$m = \frac{-8 - (-9)}{1 - 6} = \frac{1}{-5} = -\frac{1}{5}$

So, the slope of the line passing through $(6, -9)$ and $(1, -8)$ is $-\frac{1}{5}$.

Step 2: Find the slope between $(11, 4)$ and $(10, -1)$:

$m = \frac{-1 - 4}{10 - 11} = \frac{-5}{-1} = 5$

So, the slope of the line passing through $(11, 4)$ and $(10, -1)$ is $5$.

Step 3: Check if the lines are parallel, perpendicular, or intersecting.

Two lines are:

• Parallel if they have the same slope.
• Perpendicular if the product of their slopes is $-1$.

Since the slopes are $-\frac{1}{5}$ and $5$, and their product is: $-\frac{1}{5} \times 5 = -1$

The lines are perpendicular.

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Question 3

Using the equations, find the slopes and determine if the lines are parallel, perpendicular, or intersecting:

We are given:

1. $y = \frac{3}{4}x + 8$
2. $3x - 4y = 23$

Step 1: Find the slope of $y = \frac{3}{4}x + 8$.

This equation is already in slope-intercept form $y = mx + b$, where $m$ is the slope. So, the slope of the first line is: $m = \frac{3}{4}$

Step 2: Find the slope of $3x - 4y = 23$.

Rewrite this equation in slope-intercept form by solving for $y$:

$-4y = -3x + 23$ $y = \frac{3}{4}x - \frac{23}{4}$

So, the slope of the second line is also: $m = \frac{3}{4}$

Step 3: Determine if the lines are parallel, perpendicular, or intersecting.

Since both lines have the same slope $\frac{3}{4}$, they are parallel.

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Question 4

Fill out a proof given: $4(x + 3) = 20$, prove: $x = 2$:

We can solve this step-by-step as follows:

1. Given: $4(x + 3) = 20$
2. Divide both sides by 4: $x + 3 = 5$
3. Subtract 3 from both sides: $x = 2$

Thus, we have proven that $x = 2$.

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Would you like a more detailed explanation for any of these answers?

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Further Questions to Explore

1. How would you determine if two lines with unknown slopes are parallel or perpendicular?
2. If two lines have slopes $m_1$ and $m_2$, what other relationships can be inferred if $m_1 \cdot m_2 = 1$?
3. How do we interpret the y-intercept in a slope-intercept equation?
4. What does it mean for lines to intersect at an angle other than 90 degrees?
5. How can we graphically represent the solution to an equation like $4(x+3) = 20$?

Tip

For verifying parallelism or perpendicularity, always double-check the slope calculations, as small errors can affect the result.