Find k if the line through (5, 11) and (-3, 7) is parallel to the line through (-2, k) and (2, 5).

To find \( k \), we need to use the fact that the slopes of two parallel lines must be equal.

### Step 1: Calculate the slope of the line passing through \((5, 11)\) and \((-3, 7)\).

The slope formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

For the points \((5, 11)\) and \((-3, 7)\):

\[
m_1 = \frac{7 - 11}{-3 - 5} = \frac{-4}{-8} = \frac{1}{2}
\]

So, the slope of the first line is \( \frac{1}{2} \).

### Step 2: Set the slope of the second line equal to the first, since the lines are parallel.

Now, the slope of the line passing through \((-2, k)\) and \((2, 5)\) must also be \( \frac{1}{2} \). Using the slope formula for these points:

\[
m_2 = \frac{5 - k}{2 - (-2)} = \frac{5 - k}{4}
\]

Since the slopes must be equal, we set \( m_2 \) equal to \( m_1 \):

\[
\frac{5 - k}{4} = \frac{1}{2}
\]

### Step 3: Solve for \( k \).

To solve for \( k \), cross-multiply:

\[
2(5 - k) = 4(1)
\]

Simplify:

\[
10 - 2k = 4
\]

Now, solve for \( k \):

\[
10 - 4 = 2k
\]

\[
6 = 2k
\]

\[
k = 3
\]

### Final Answer:
The value of \( k \) is \( 3 \).

Would you like further details or have any questions?

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Here are 5 related questions to deepen your understanding:
1. How do you find the equation of a line given two points?
2. What happens to the slopes of two perpendicular lines?
3. How do you find the y-intercept once you have the slope and a point?
4. How do you determine if three points lie on the same line?
5. What is the geometric interpretation of a slope of zero?

**Tip:** For any two points, the slope formula is the ratio of the change in y-coordinates to the change in x-coordinates. This gives a precise measure of how steep a line is!

Learn how to find the value of k when given two lines passing through points (5, 11), (-3, 7), (-2, k), and (2, 5), which are parallel. This problem covers key concepts in algebra and coordinate geometry, including the slope formula and properties of parallel lines. Suitable for students in Grades 8-10.

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