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

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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.

Answer 1

To solve this problem, let's break it down step by step.

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

To find the slope, we can use the formula for slope \( m \) which is given by:

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

Where:

• \( y_1 = 34 \) (water level after 3 days)
• \( y_2 = 62 \) (water level after 10 days)
• \( x_1 = 3 \) (days)
• \( x_2 = 10 \) (days)

Now we calculate:

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

Response for part a: 4

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

The y-intercept (b) represents the water level before the rain started, which is given as \( 22 \) feet.

Response for part b: 22

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

The slope-intercept form of a linear equation is given by:

\[ y = mx + b \]

Where \( m \) is the slope and \( b \) is the y-intercept:

Using the values we've found:

• \( m = 4 \)
• \( b = 22 \)

The equation is:

\[ y = 4x + 22 \]

Response for part c: y = 4x + 22

d. Write this equation in standard form.

The standard form of a linear equation is typically written as:

\[ Ax + By = C \]

Starting from our slope-intercept form \( y = 4x + 22 \), we can rearrange it:

1. Subtract \( 4x \) from both sides:

\[ -4x + y = 22 \]

2. Multiply through by -1 to make the x-coefficient positive:

\[ 4x - y = -22 \]

Thus, the equation in standard form is:

Response for part d: 4x - y = -22

Answer 2

for part d, that is not an option, try again.

Answer 3

Let's reevaluate part d and ensure we express it correctly in standard form.

Starting from the slope-intercept form:

\[ y = 4x + 22 \]

To rewrite it in standard form \( Ax + By = C \), we rearrange the equation as follows:

1. Move \( 4x \) to the left side of the equation:

\[ -4x + y = 22 \]

2. To express it in the traditional \( Ax + By = C \) format with \( A \) being positive, we can multiply the entire equation by -1:

\[ 4x - y = -22 \]

However, the standard form usually has positive coefficients for \( A \). Therefore, the correct form will be:

\[ 4x - y = -22 \]

If you need to express the equation in a more conventional format where all coefficients are positive, we can multiply by -1 again:

\[ -4x + y = 22 \]

But typically, we stick with the version where \( A \) is positive, which leads us back to:

Response for part d: 4x - y = -22

If this is not an acceptable form, could you please specify the expected format for the standard form?

Answer 4

what is another form of 4x - y = -22

Answer 5

The equation \( 4x - y = -22 \) can be rearranged into several equivalent forms. One common form is to manipulate it to express \( y \) in terms of \( x \), which would put it back into slope-intercept form. However, if you're looking for a different representation while still keeping it in a standard or general linear form, we could move it around to express it in terms of \( C \), while keeping its structure.

Here is one alternative way to express the equation:

Rearranging to y = mx + b form

From \( 4x - y = -22 \), you can solve for \( y \):

1. Add \( y \) to both sides and then add \( 22 \) to both sides:

\[ 4x + 22 = y \]

Rearranging gives:

\[ y = 4x + 22 \]

Alternate standard form

You could also rearrange \( 4x - y = -22 \) simply by moving terms around:

\[ -4x + y = 22 \]

Writing it in a different standard form

If you want to express as \( Ax + By = C \) with all positive coefficients, you can multiply the entire equation \( 4x - y = -22 \) by -1:

\[ -4x + y = 22 \]

This is not strictly different in meaning, but it does change the coefficients.

In summary, possible forms include:

1. \( 4x - y = -22 \) (original)
2. \( y = 4x + 22 \) (slope-intercept form)
3. \( -4x + y = 22 \) (all positive coefficients)

If you are looking for even another specific transformation, please let me know!