Question

Suppose that the water level of a river is 34 feet after 3 days of rain and 62 feet after 10 days of rain. The river was 22 feet before the rain began.(4 points)
Put responses in the correct response input area to answer the question. Select a response, navigate to the desired input area and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
a. What is the rate of change (slope) for this scenario? Press enter key to drop response here.Response input area

b. What is the y-intercept for this scenario? Press enter key to drop response here.Response input area

c. Write an equation for the scenario in slope-intercept form. Press enter key to drop response here.Response input area

d. Write this equation in standard form. Press enter key to drop response here.Response input area

Answer 1

To solve this problem, we need to find the rate of change (slope) and the intercepts based on the data provided about the river's water levels over time.

### a. What is the rate of change (slope) for this scenario?

To find the slope, we use the formula:
\[
\text{slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let's take the two points from the information given:
- Day 3: Water level = 34 feet
- Day 10: Water level = 62 feet

Thus, our points are:
- \( (3, 34) \)
- \( (10, 62) \)

Now plug the values into the slope formula:
\[
\text{slope} = \frac{62 - 34}{10 - 3} = \frac{28}{7} = 4
\]

**The rate of change (slope) is 4.**

### Response for (a):
4

---

### b. What is the y-intercept for this scenario?

To find the y-intercept, we will use the slope-intercept form of a linear equation, which is:
\[
y = mx + b
\]
where \(m\) is the slope and \(b\) is the y-intercept. We will use one of our points and the slope we calculated.

Using the point (3, 34):

\[
34 = 4(3) + b
\]
\[
34 = 12 + b
\]
\[
b = 34 - 12 = 22
\]

**The y-intercept is 22.**

### Response for (b):
22

---

### c. Write an equation for the scenario in slope-intercept form.

Using the slope \(m = 4\) and the y-intercept \(b = 22\), we can write the equation in slope-intercept form:

\[
y = 4x + 22
\]

### Response for (c):
y = 4x + 22

---

### d. Write this equation in standard form.

The standard form of a linear equation is given by:
\[
Ax + By = C
\]
To convert \(y = 4x + 22\) to standard form, we need to rearrange it:

\[
-4x + y = 22
\]
Multiplying through by -1 for a standard convention (if necessary):
\[
4x - y = -22
\]

Now, to keep the coefficients as integers, we adjust this:

\[
4x - y = 22
\]

### Response for (d):
4x - y = 22