Question

Gina charges an initial fee and an hourly fee to babysit. Using the table below find the hourly fee and the initial fee that

Gina charges to babysit. Show work.

Answer 1

By analyzing this graph we can see that it is a linear relation. Meaning that the Rate of Change is a consistent number, in this case $7. We know its $7 because by looking at the y coordinate for all present coordinates we can see a consistent difference of $7. From this point you could work backwards. Finding the coordinates for the first hour, which by doing $26 - $7 tells us the missing number for the first coordinate pair (1,19). Then we work backwards one more time, this time subtracting $7 from $19, and this tells us the initial value (the coordinate on the y-axis), which is $12. One thing you can do to show your work is draw a straight line that intersects all the coordinates and the y-axis, the line should intersect the y-axis at these coordinates (0,12). Now I could be wrong but if there is a formula that is used to determine this answer then I am unaware of it. Hope this helps however

Answer 2

Gina charges $
`7` per hour for babysitting with an initial fee of $
`12`.

To find the initial fee and hourly fee, you can use the information provided in the table to set up a system of equations.

Let h be the hourly fee and i be the initial fee.

Given data points:

`\text{Hours (x)} & : 2, 3, 4, 5 \\\text{Cost (y)} & : 26, 33, 40, 47`

The general formula for the cost y in terms of x, h, and i is:

`\[ y = h \cdot x + i \]`

Now, substitute the values from the table into this formula to create a system of equations:

1. For
`\( x = 2 \), \( y = 26 \):\[ 26 = 2h + i \]`

2. For
`\( x = 3 \), \( y = 33 \):\[ 33 = 3h + i \]`

3. For
`\( x = 4 \), \( y = 40 \):\[ 40 = 4h + i \]`

4. For
`\( x = 5 \), \( y = 47 \):\[ 47 = 5h + i`

Now, you have a system of four equations:

`1. \quad 26 &= 2h + i \\2. \quad 33 &= 3h + i \\3. \quad 40 &= 4h + i \\4. \quad 47 &= 5h + i \\`

Here's one way to solve it:

Subtract equation (
`1`) from equation (
`2`):

`\[ (2) - (1) \implies 33 - 26 = (3h + i) - (2h + i) \]\[ 7 = h \]`

Now that we know h =
`7`, substitute this value into any of the original equations. Let's use equation (
`1`):

`\[ 26 = 2(7) + i \]\[ 26 = 14 + i \]\[ i = 26 - 14 \]\[ i = 12 \]`

So, the solution to the system is
`\( h = 7 \)` (hourly fee) and
`\( i = 12 \)` (initial fee). Therefore, Gina charges $
`7` per hour for babysitting, and there is an initial fee of $
`12`.