​f(x)equals5 x minus 7.
​(a)  Find the average rate of change from 5 to 8.
​(b)  Find an equation of the secant line containing left parenthesis 5 comma f left parenthesis 5 right parenthesis right parenthesis and left parenthesis 8 comma f left parenthesis 8 right parenthesis right parenthesis .

Given the function \( f(x) = 5x - 7 \), we are tasked with finding:

### (a) The average rate of change from \( x = 5 \) to \( x = 8 \)

The formula for the average rate of change of a function between two points \( x = a \) and \( x = b \) is:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

Here, \( a = 5 \) and \( b = 8 \). First, we need to calculate \( f(5) \) and \( f(8) \):

- \( f(5) = 5(5) - 7 = 25 - 7 = 18 \)
- \( f(8) = 5(8) - 7 = 40 - 7 = 33 \)

Now, using the formula:

\[
\text{Average Rate of Change} = \frac{f(8) - f(5)}{8 - 5} = \frac{33 - 18}{3} = \frac{15}{3} = 5
\]

So, the average rate of change is \( 5 \).

### (b) Equation of the secant line through \( (5, f(5)) \) and \( (8, f(8)) \)

The secant line is a straight line passing through the two points \( (5, f(5)) \) and \( (8, f(8)) \). These points are:

- \( (5, f(5)) = (5, 18) \)
- \( (8, f(8)) = (8, 33) \)

To find the equation of the secant line, we use the point-slope form of the equation of a line:

\[
y - y_1 = m(x - x_1)
\]

Where \( m \) is the slope (which is the average rate of change calculated earlier), and \( (x_1, y_1) \) is one of the points on the line (we can use either \( (5, 18) \) or \( (8, 33) \)).

We already know that the slope \( m = 5 \), so using the point \( (5, 18) \):

\[
y - 18 = 5(x - 5)
\]

Simplifying:

\[
y - 18 = 5x - 25
\]
\[
y = 5x - 25 + 18
\]
\[
y = 5x - 7
\]

Thus, the equation of the secant line is \( y = 5x - 7 \).

### Summary:
- **(a)** The average rate of change is \( 5 \).
- **(b)** The equation of the secant line is \( y = 5x - 7 \).

Would you like more details or have any questions?

Here are some related questions:
1. What is the significance of the average rate of change for linear functions?
2. How does the secant line differ from a tangent line in calculus?
3. What happens if we compute the average rate of change over a larger interval for nonlinear functions?
4. How would the process change if the function was quadratic?
5. What is the geometric interpretation of the average rate of change?

**Tip**: For linear functions, the secant line between any two points will always be the same as the line itself!

f(x) equals 5x minus 7. (a) Find the average rate of change from 5 to 8. (b) Find an equation of the secant line containing (5, f(5)) and (8, f(8)).

Explore how to calculate the average rate of change and find the equation of the secant line for the linear function f(x) = 5x - 7, between the points x = 5 and x = 8. This step-by-step solution uses key algebraic formulas suitable for students in Grades 9-10.

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