Question

A pet sitter charges 25$ per day and an additional $5 for each additional overnight stay. Write an equation in standard form to represent this situation.

Answer 1

Identify the equation that passes through the point (-2,5) and is perpendicular to the line that passes through (-1,-4) and (1,4).

Answer 2

In point-slope form, please.

Answer 3

Let \( d \) represent the number of days the pet sitter is hired, and \( n \) represent the number of additional overnight stays.

The total cost \( C \) for the pet sitting services can be expressed as:

\[
C = 25d + 5n
\]

To write this equation in standard form, we typically rearrange it into the form \( Ax + By = C \).

Here, we can represent the costs as follows:

\[
-25d - 5n + C = 0
\]

To convert this to standard form \( Ax + By = C \), let's rewrite the equation:

\[
25d + 5n = C
\]

Where \( A = 25 \), \( B = 5 \), and \( C \) is the total cost.

Thus, the equation in standard form is:

\[
25d + 5n = C
\]

This represents the total cost \( C \) based on the number of days \( d \) and additional overnight stays \( n \).

Answer 4

To find the equation of the line that passes through the point \((-2, 5)\) and is perpendicular to the line that passes through the points \((-1, -4)\) and \((1, 4)\), we can follow these steps:

1. **Find the slope of the line through the points \((-1, -4)\) and \((1, 4)\)**.

The slope \( m \) of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:

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

Substituting the points \((-1, -4)\) and \((1, 4)\):

\[
m = \frac{4 - (-4)}{1 - (-1)} = \frac{4 + 4}{1 + 1} = \frac{8}{2} = 4
\]

So, the slope of the line through these points is \(4\).

2. **Find the slope of the line that is perpendicular**.

The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the first line. The negative reciprocal of \(4\) is:

\[
m_{\text{perpendicular}} = -\frac{1}{4}
\]

3. **Use the point-slope form of the equation of a line**.

The point-slope form of a line's equation is given by:

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

Where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. We have the point \((-2, 5)\) and the slope \(-\frac{1}{4}\):

\[
y - 5 = -\frac{1}{4}(x - (-2))
\]

Simplifying this:

\[
y - 5 = -\frac{1}{4}(x + 2)
\]

Distributing \(-\frac{1}{4}\):

\[
y - 5 = -\frac{1}{4}x - \frac{2}{4}
\]
\[
y - 5 = -\frac{1}{4}x - \frac{1}{2}
\]

Adding \(5\) to both sides to solve for \(y\):

\[
y = -\frac{1}{4}x - \frac{1}{2} + 5
\]
\[
y = -\frac{1}{4}x + \frac{10}{2} - \frac{1}{2}
\]
\[
y = -\frac{1}{4}x + \frac{9}{2}
\]

4. **Convert to slope-intercept form or standard form**.

To express this in standard form \(Ax + By = C\):

Multiply through by 4 to eliminate the fraction:

\[
4y = -x + 18
\]

Rearranging gives:

\[
x + 4y = 18
\]

Thus, the equation of the line that passes through the point \((-2, 5)\) and is perpendicular to the line that goes through \((-1, -4)\) and \((1, 4)\) is:

\[
\boxed{x + 4y = 18}
\]

Answer 5

Sure! The point-slope form of a linear equation is given by:

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

where \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope.

We previously calculated the slope of the line that is perpendicular to the line through the points \((-1, -4)\) and \((1, 4)\) to be:

\[
m = -\frac{1}{4}
\]

We are also given that the line passes through the point \((-2, 5)\).

Now, using the point-slope form with \( (x_1, y_1) = (-2, 5) \) and \( m = -\frac{1}{4} \):

\[
y - 5 = -\frac{1}{4}(x - (-2))
\]

This can be simplified to:

\[
y - 5 = -\frac{1}{4}(x + 2)
\]

Thus, the equation of the line in point-slope form is:

\[
\boxed{y - 5 = -\frac{1}{4}(x + 2)}
\]