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)

Question

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

b. What is the y-intercept for this scenario? (Response area)

c. Write an equation for the scenario in slope-intercept form. (Response area)

d. Write this equation in standard form. (response area)

responses
28
22
4
(0, 34)
y = 4x - 22
-4x + y = 22
-4x + y = 34
7
0
y = 4x + 34
4x + = 34
(0, 0)
(0, 22)
y = 4x + 22
y = 28x + 34
(0, 4)

Answer 1

To solve this problem, let’s analyze the information given.

Initial Data:

• Water level is 22 feet before the rain starts (at day 0).
• Water level is 34 feet after 3 days of rain.
• Water level is 62 feet after 10 days of rain.

Now, let’s perform the tasks step by step.

a. Find the rate of change (slope).

Using the two data points:

1. After 3 days: (3, 34)
2. After 10 days: (10, 62)

The formula for slope (rate of change) is given by: \[ \text{slope} (m) = \frac{y_2 - y_1}{x_2 - x_1} \] Where \((x_1, y_1) = (3, 34)\) and \((x_2, y_2) = (10, 62)\).

So: \[ m = \frac{62 - 34}{10 - 3} = \frac{28}{7} = 4 \]

b. Find the y-intercept.

The y-intercept is the water level when \(x = 0\) (before the rain starts). From the problem statement, we know the water level before rain began was: \[ \text{y-intercept} = 22 \]

c. Write an equation in slope-intercept form (y = mx + b).

Using the slope \(m = 4\) and the y-intercept \(b = 22\): \[ y = 4x + 22 \]

d. Write the equation in standard form.

Standard form is typically expressed as \(Ax + By = C\). Starting with the slope-intercept form: \[ y = 4x + 22 \] We can rearrange this: \[ -4x + y = 22 \]

To commonly express this in standard form: \[ 4x - y = -22 \] However, standard form traditionally keeps \(A\) as a non-negative integer, so we can multiply through by -1: \[ -4x + y = 22 \]

Final Responses:

• a: 4 (rate of change)
• b: 22 (y-intercept)
• c: \(y = 4x + 22\) (slope-intercept form)
• d: \(-4x + y = 22\) (standard form)

Answer 2

Suppose that the water level of a river is 34 feet after 3 days of rain and 62 feet after 10 days of rain.

Given the two points we know: (3, 34) and (10, 62), what would the point-slope equation look like?

(1 point)
Responses

(y - 34) = 4 (x - 10)
(y - 34) = 4 (x - 10)

(y - 10) = 4 (x + 62)
(y - 10) = 4 (x + 62)

(y - 34) = 4 (x - 3)
(y - 34) = 4 (x - 3)

(y - 3) = 4 (x - 34)

Answer 3

To write a point-slope equation, we use the formula:

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

where \(m\) is the slope and \((x_1, y_1)\) is one of the points on the line.

From the previous calculation, we found:

• The slope \(m = 4\).
• We can choose either of the points \((3, 34)\) or \((10, 62)\).

Using the point \((3, 34)\): \[ y - 34 = 4(x - 3) \]

Alternatively, using the point \((10, 62)\): \[ y - 62 = 4(x - 10) \]

From the list of responses provided, the correct option that matches our derived point-slope equation using the point \((3, 34)\) is:

(y - 34) = 4 (x - 3)

So, the answer is: (y - 34) = 4 (x - 3).

Answer 4

Solve this equation for y: 4y=3x
(1 point)
Responses

y = 12x
y = 12x

y = 4/3 x
y = 4/3 x

y = 2x -4
y = 2x -4

y = ¾ x

Answer 5

To solve the equation \(4y = 3x\) for \(y\), we need to isolate \(y\) by dividing both sides of the equation by 4:

\[ y = \frac{3x}{4} \]

This can also be written as:

\[ y = \frac{3}{4}x \]

From the options given, the correct response is:

y = ¾ x.

Answer 6

Solve the equation for x: 2x+b=c
Check this process:

Step 1: 2x+b−b=c−b
Step 2: 2x=c−b
Step 3: 2x−2=c−b−2
Step 4: x=c2−b−2
Check all of the choices that are true statements.

(2 points)
Responses

There is no error in the process.
There is no error in the process.

The process has an error from step 1 to step 2
The process has an error from step 1 to step 2

The process has an error from step 2 to step 3
The process has an error from step 2 to step 3

The process has an error from step 3 to step 4

Answer 7

Let's review the process step by step.

The original equation is: \[ 2x + b = c \]

Step 1: \(2x + b - b = c - b\)
This step is correct. It effectively subtracts \(b\) from both sides.

Step 2: \(2x = c - b\)
This step is also correct, as it simplifies the equation by removing \(b\) from the left side.

Step 3: \(2x - 2 = c - b - 2\)
This step contains an error. The equation should remain as \(2x = c - b\). You cannot just subtract 2 from both sides as done here without justification. Thus, there is an error in this step.

Step 4: \(x = \frac{c - 2 - b}{2}\)
This step also contains an error because it is based on the incorrect step 3 formula. The correct next step would involve dividing \(c - b\) by 2 to isolate \(x\): \[ x = \frac{c - b}{2} \]

Summary:

• Step 1 to Step 2: No error.
• Step 2 to Step 3: Error.
• Step 3 to Step 4: Error.

True Statements:

1. There is no error in the process. (False)
2. The process has an error from step 1 to step 2. (False)
3. The process has an error from step 2 to step 3. (True)
4. The process has an error from step 3 to step 4. (True)

Thus, the correct statements to check are:

• The process has an error from step 2 to step 3.
• The process has an error from step 3 to step 4.

Answer 8

The equation of this graph is y=23x+2

What transformation would need to happen for the equation of the line to be y=23x
?

(1 point)
Responses

Shift the graph right 3, down 2
Shift the graph right 3, down 2

Shift the graph left 2
Shift the graph left 2

Shift the graph down 3, left 2
Shift the graph down 3, left 2

Shift the graph down 2

Answer 9

The equation of the original line is given as:

\[ y = 23x + 2 \]

The equation of the modified line is:

\[ y = 23x \]

To find the transformation that changes the first equation to the second, we observe the change in the constant term.

The original equation includes \(+2\), and the new equation has \(0\) (since \(y = 23x\) can be seen as \(y = 23x + 0\)).

To transition from \(y = 23x + 2\) to \(y = 23x\), we need to shift the graph down by 2 units.

Therefore, the correct response is:

Shift the graph down 2.

Answer 10

Given the table for x, time in minutes, and y, distance traveled:

x: time 8 5 7 2 9
y: distance 664 415 581 166 747


(4 points)
b. Write an equation to show the relationship between x and y: (response area)
.

c. This relationship is (response area) because the y-intercept is (response area)

B responses
12y = x
y = x - 83
y = 12x
y = 83x

C responses
proportional
non-proportional
(0, 0)
(8, 664)
(0, 83

Answer 11

To find the equation that relates \(x\) (time in minutes) and \(y\) (distance traveled), let's analyze the data provided. First, we need to derive the slope of the relationship, if it's linear.

Using two points from the table, for example, \( (8, 664) \) and \( (2, 166) \):

Step 1: Calculate the slope (m)

The slope (m) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Let's pick \( (8, 664) \) as \( (x_1, y_1) \) and \( (2, 166) \) as \( (x_2, y_2) \):

\[ m = \frac{166 - 664}{2 - 8} = \frac{-498}{-6} = 83 \]

Step 2: Write the equation in slope-intercept form

Using the slope \(m = 83\) and the point \( (8,664) \), we can use the point-slope form: \[ y - y_1 = m(x - x_1) \] Substituting the values: \[ y - 664 = 83(x - 8) \] Now, let's simplify it to slope-intercept (y = mx + b) form:

1. Expand: \[ y - 664 = 83x - 664 \]
2. Add 664 to both sides: \[ y = 83x \]

Answer to part b

The equation showing the relationship between \(x\) and \(y\) is: y = 83x

Answer to part c

Now, examining the relationship:

• This relationship is proportional because it can be described by a linear equation in the form \(y = kx\) with no constant, meaning that when \(x = 0\), \(y = 0\).

Completion of responses

• For part b, the equation is: y = 83x

• For part c, the relationship is: proportional

• The y-intercept is: (0, 0)

So, to summarize:

• b. y = 83x
• c. proportional, and the y-intercept is (0, 0).