find k so that the line through (2,-1) and (k,4) is parallel to 3x+2y=4 and perpendicular to 4x-3y=-1 and what is k

Let's solve this problem step by step.

### Step 1: Equation of the line parallel to \( 3x + 2y = 4 \)
The slope of a line in the form \( Ax + By = C \) is given by \( -\frac{A}{B} \).

For the line \( 3x + 2y = 4 \), the slope is:
\[
m_1 = -\frac{3}{2}
\]
So, the line through \( (2, -1) \) and \( (k, 4) \) must have the same slope because it is parallel.

### Step 2: Find the slope between points \( (2, -1) \) and \( (k, 4) \)
The slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the points \( (2, -1) \) and \( (k, 4) \), we get:
\[
m = \frac{4 - (-1)}{k - 2} = \frac{5}{k - 2}
\]
Since this slope must be equal to \( -\frac{3}{2} \), we set up the equation:
\[
\frac{5}{k - 2} = -\frac{3}{2}
\]

### Step 3: Solve for \( k \)
To solve for \( k \), cross-multiply:
\[
5 \cdot 2 = -3 \cdot (k - 2)
\]
\[
10 = -3k + 6
\]
Now solve for \( k \):
\[
10 - 6 = -3k
\]
\[
4 = -3k
\]
\[
k = -\frac{4}{3}
\]

Thus, the value of \( k \) is \( -\frac{4}{3} \).

---

Would you like me to verify this solution further, or clarify any steps?

Here are some related questions:
1. What is the equation of a line parallel to \( 3x + 2y = 4 \) through a different point?
2. How do you derive the general form of a line given a slope and a point?
3. What if the line through \( (2, -1) \) and \( (k, 4) \) was perpendicular to \( 3x + 2y = 4 \)?
4. How would the slope change if you had the equation \( 2x + 3y = 6 \)?
5. Can you solve for \( k \) when a line is parallel to the y-axis?

**Tip:** To determine if two lines are parallel, always compare their slopes!

Find k so that the line through (2,-1) and (k,4) is parallel to 3x+2y=4 and perpendicular to 4x-3y=-1. What is k?

Learn how to find the value of k such that the line through (2, -1) and (k, 4) is parallel to 3x + 2y = 4 and perpendicular to 4x - 3y = -1. This problem covers key concepts in algebra and coordinate geometry, including slopes, parallel lines, and perpendicular lines, with detailed step-by-step solutions. Suitable for Grades 9-12.

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