Find a linear function​ h, given ​h(3​)equalsnegative 5 and ​h(negative 4​)equals23. Then find ​h(5​).
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Part 1
h left parenthesis x right parenthesisequals

enter your response here
​(Type an expression using x as the variable. Simplify your​ answer.)

To find a linear function \( h(x) = mx + b \), we need to determine the slope \( m \) and the y-intercept \( b \).

We are given two points: 
\( h(3) = -5 \) and \( h(-4) = 23 \). These points can be written as \( (3, -5) \) and \( (-4, 23) \).

### Step 1: Find the slope \( m \)

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

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

Substitute the given points \((3, -5)\) and \((-4, 23)\):

\[
m = \frac{23 - (-5)}{-4 - 3} = \frac{23 + 5}{-4 - 3} = \frac{28}{-7} = -4
\]

### Step 2: Find the y-intercept \( b \)

Using the slope \( m = -4 \) and one of the points, say \( (3, -5) \), we can plug these into the linear equation \( h(x) = mx + b \) to solve for \( b \).

\[
-5 = -4(3) + b
\]
\[
-5 = -12 + b
\]
\[
b = -5 + 12 = 7
\]

### Step 3: Write the equation of the linear function

Now that we have the slope \( m = -4 \) and the y-intercept \( b = 7 \), the equation of the linear function is:

\[
h(x) = -4x + 7
\]

### Part 2: Find \( h(5) \)

Now, substitute \( x = 5 \) into the linear function \( h(x) = -4x + 7 \):

\[
h(5) = -4(5) + 7 = -20 + 7 = -13
\]

### Final Answer:
- The linear function is \( h(x) = -4x + 7 \).
- \( h(5) = -13 \).

Would you like more details on any part of the solution or have additional questions?

Here are some related questions to explore:
1. How do you find the slope from any two points on a line?
2. How do changes in the slope affect the graph of a linear function?
3. How can we graph a linear function step-by-step?
4. What is the significance of the y-intercept in a linear equation?
5. How can we quickly verify if a point lies on the graph of a given linear function?

**Tip:** Always simplify slope and intercept calculations to avoid errors in finding the final function.

Find a linear function h, given h(3) = -5 and h(-4) = 23. Then find h(5).

Learn how to find the equation of a linear function h(x) = -4x + 7 given two points, h(3) = -5 and h(-4) = 23, and evaluate h(5). This problem covers algebraic concepts including slope-intercept form and linear equations. Suitable for students in Grades 8-10.

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