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We are given two equations of lines:

1. \( y = 3x - 5 \)
2. \( y = -2x + 10 \)

The lines intersect at the point \( (h, k) \). To find the value of \( k \), we need to determine the \( y \)-coordinate at the intersection point. First, we will solve the system of equations by setting them equal to each other, since at the point of intersection, both equations share the same \( y \)-value.

### Step 1: Set the equations equal to each other
\[
3x - 5 = -2x + 10
\]

### Step 2: Solve for \( x \)
Add \( 2x \) to both sides:
\[
3x + 2x - 5 = 10
\]
\[
5x - 5 = 10
\]
Now add 5 to both sides:
\[
5x = 15
\]
Divide by 5:
\[
x = 3
\]

### Step 3: Find \( y \) (or \( k \))
Now that we know \( x = 3 \), substitute this value into either of the original equations to find \( y \). Let's use the first equation:

\[
y = 3x - 5
\]
Substitute \( x = 3 \):
\[
y = 3(3) - 5 = 9 - 5 = 4
\]

Thus, \( k = 4 \).

### Final Answer:
The value of \( k \) is \( 4 \).

---

Would you like more details or have any questions about the solution?

Here are 5 related questions you might find helpful:
1. What is the value of \( h \) in this example?
2. How do you find the point of intersection of two lines algebraically?
3. What does it mean geometrically for two lines to intersect?
4. Can two lines with the same slope have an intersection point?
5. How would the problem change if the lines were parallel?

**Tip:** Always check your solution by substituting both values back into the original equations to verify the intersection point!

If the lines y = 3x - 5 and y = -2x + 10 intersect at the point (h, k), what is the value of k?

Learn how to find the intersection point of two lines y = 3x - 5 and y = -2x + 10 by solving the system of equations. This example demonstrates solving for both the x and y coordinates (h, k), focusing on key algebraic concepts such as linear equations and system solving.

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