+2441 1. Days
The number of days that pass cannot be controlled as one cannot simply stop or speed up time, so this is the independent variable.
2. Pounds of Dog Food
The amount of pounds of dog food can be controlled by consuming less per day, so this is the dependent variable. Also, the number of days that pass linearly influences the amount of pounds remaining.
3.
There are a few methods to calculate the slope here, but I will simply select two points on the provided graph, which will be \((30,0)\) and \((0,40)\). Use the slope formula to calculate the slope.
\(m=\frac{0-40}{30-0}=\frac{-40}{30}=-\frac{4}{3}\)
This is the slope.
4.
The y-intercept of this function is located at \((0,40)\). For this problem, this indicates the number of pounds of dog food originally in the bag before consumption.
5.
We already know the slope, and we already know the y-intercept, so it is relatively simple to craft the function using function notation. Just put the pieces together.
Let d = number of days
\(f(d)=-\frac{4}{3}d+40\)
6. To rewrite from function notation to slope-intercept notation, simply replace the \(f(d)\) with a y.
\(y=-\frac{4}{3}x+40\)
7.
The standard form of a line is written in the form \(Ax+By=C\) such that A,B, and C are all integers, A is equal to or greater than 0. Simply rearrange the equation written in slope-intercept form.
| \(y=-\frac{4}{3}x+40\) | Add 4/3x to both sides. |
| \(\frac{4}{3}x+y=40\) | Multiply by 3 on both sides to eliminate the fraction. |
| \(4x+3y=120\) | This equation meets all the guidelines for an equation in standard form. |
+2441 Of course! I will gladly delve deeper into this subject.
The explanation I provided is probably unsatisfactory and shoddy anyway. I believe that these rules are best demonstrated by example.
1. Divisibility by 7
Is \(205226\) divisible by 7? Well, let's use the process!
| \(\textcolor{blue}{20522}\textcolor{red}{6}\) | |
| 1. \(\textcolor{blue}{20522}-2*\textcolor{red}{6}=\textcolor{blue}{2051}\textcolor{red}{0}\) | I have no clue if this is indeed divisible by 7, so do this process again and again (hence recursion) |
| 2. \(\textcolor{blue}{2051}-2*\textcolor{red}{0}=\textcolor{blue}{205}\textcolor{red}{1}\) | I still cannot tell, so I will do this again. |
| 3. \(\textcolor{blue}{205}-2*\textcolor{red}{1}=\textcolor{blue}{20}\textcolor{red}{3}\) | I still cannot tell. |
| 4. \(\textcolor{blue}{20}-2*\textcolor{red}{3}=\textcolor{blue}{1}\textcolor{red}{4}\) | I know that 14 is divisible by 7, so the original number is, too. |
How about \(22604\)? Well, let's check it!
| \(\textcolor{blue}{2260}\textcolor{red}{4}\) | |
| 1. \(\textcolor{blue}{2260}-2*\textcolor{red}{4}=\textcolor{blue}{225}\textcolor{red}{2}\) | Yet again, I cannot make a judgment. |
| 2. \(\textcolor{blue}{225}-2*\textcolor{red}{2}=\textcolor{blue}{22}\textcolor{red}{1}\) | Of course, we must keep going. |
| 3. \(\textcolor{blue}{22}-2*\textcolor{red}{1}=20\) | I know that this number is not divisible by 7, so the original number is not either. |
2. Divisibility by 11
Let's check if \(43923\) is divisible.
| \(\textcolor{blue}{4392}\textcolor{red}{3}\) | |
| 1. \(\textcolor{blue}{4392}-\textcolor{red}{3}=\textcolor{blue}{438}\textcolor{red}{9}\) | Let's do it again! |
| 2. \(\textcolor{blue}{438}-\textcolor{red}{9}=\textcolor{blue}{42}\textcolor{red}{9}\) | One more time! |
| \(\textcolor{blue}{42}-\textcolor{red}{9}=\textcolor{blue}{3}\textcolor{red}{3}\) | I know that 33 is divisible by 11, so the original number is, too. |
How about \(123567\)?
| \(\textcolor{blue}{12356}\textcolor{red}{7}\) | |
| 1. \(\textcolor{blue}{12356}-\textcolor{red}{7}=\textcolor{blue}{1234}\textcolor{red}{9}\) | This requires perserverance. Keep going! |
| 2. \(\textcolor{blue}{1234}-\textcolor{red}{9}=\textcolor{blue}{122}\textcolor{red}{5}\) | |
| 3. \(\textcolor{blue}{122}-\textcolor{red}{5}=\textcolor{blue}{11}\textcolor{red}{7}\) | I know that \(11*11=121\), so 117 is not divisible. |
3. Divisibility by 13
Is \(19704\) divisible? Let's find out!
| \(\textcolor{blue}{1970}\textcolor{red}{4}\) | |
| 1. \(\textcolor{blue}{1970}+4*\textcolor{red}{4}=\textcolor{blue}{198}\textcolor{red}{6}\) | |
| 2. \(\textcolor{blue}{198}+4*\textcolor{red}{6}=\textcolor{blue}{22}\textcolor{red}{2}\) | |
| \(=\textcolor{blue}{22}+4*\textcolor{red}{2}=\textcolor{blue}{3}\textcolor{red}{0}\) | \(13*3=36\), so 30 is not divisible by 13 and nor is the given number. |
Is \(9321 \) able to be divised?
| \(\textcolor{blue}{932}\textcolor{red}{1}\) | |
| 1. \(\textcolor{blue}{932}+4*\textcolor{red}{1}=\textcolor{blue}{93}\textcolor{red}{6}\) | |
| 2. \(\textcolor{blue}{93}+4*\textcolor{red}{6}=\textcolor{blue}{11}\textcolor{red}{7}\) | |
| 3. \(\textcolor{blue}{11}+4*\textcolor{red}{7}=\textcolor{blue}{3}\textcolor{red}{9}\) | 39 is divisible by 13, so the original number is as well. |
+2441 | Divisibility Test Number | Process |
| 1 | Every integer is divisible by 1 |
| 2 | Check to see if the last digit is divisible by 2 |
| 3 | Recursively check if the sum of digits is divisible by 3 |
| 4 | Check to see if the last two digits are divisible by 4 |
| 5 | Check to see if the last digit is divisible by 5 |
| 6 | Check divisibility rules for 2 & 3 |
| 7 | Recursively check if subtracting twice the sum of the last digit from the rest of the number is divisible by 7 |
| 8 | Check if the last 3 digits are divisible by 8 |
| 9 | Recursively check if the sum of the digits is divisible by 9 |
| 10 | Check if the last digit is a 0 |
| 11 | Recursively check if subtracting the final digit from the rest is divisible by 11 |
| 12 | Check divisibility for 3 & 4 |
| 13 | Recursively add 4 times the final digit to the rest |
+2441 Use the midpoint formula to get you started and then work from there!
The midpoint formula tells you to find the average of the x-coordinates and y-coordinates.
| \((\frac{x-9}{2},\frac{y+1}{2})\) | Now that we have the expression for both the x- and y-coordinate, solve for the individual variables. | ||
| Multiply both equations by 2 to eliminate the fraction. | ||
| Subtract the constant to isolate the variable. | ||
| |||
| \((15,-11)\) | This is the coordinate. | ||
+2441 1)
\(\frac{\textcolor{red}{\frac{3}{x}+\frac{1}{4}}}{\textcolor{blue}{1+\frac{3}{x}}}\) is quite the complex fraction. First, let's just deal with the numerator
| \(\textcolor{red}{\frac{3}{x}+\frac{1}{4}}\) | First. let's change this into fractions with common denominators. The LCM is 4x. |
| \(\frac{12}{4x}+\frac{x}{4x}\) | Now, combine the fractions because we have formed a common denominator. |
| \(\frac{12+x}{4x}\) | |
| \(\textcolor{blue}{1+\frac{3}{x}}\) | Just like before, transform the fractions to create a common denominator and combine. |
| \(\frac{x}{x}+\frac{3}{x}\) | |
| \(\frac{x+3}{x}\) | |
Now, write it as a fraction; you'll see how much easier it is to work with!
| \(\frac{\textcolor{red}{\frac{3}{x}+\frac{1}{4}}}{\textcolor{blue}{1+\frac{3}{x}}}=\frac{\frac{12+x}{4x}}{\frac{x+3}{x}}\) | |
| \(\frac{\frac{12+x}{4x}}{\frac{x+3}{x}}\) | Multiply by \(\frac{x}{x+3}\), the reciprocal of the complex denominator, to eliminate this complex fraction. |
| \(\frac{x(12+x)}{4x(x+3)}\) | The x in the numerator and the x in the denominator cancel out here. |
| \(\frac{12+x}{4(x+3)}\) | Now, distribute the 4 into every term. |
| \(\frac{12+x}{4x+12}\) | |
2) \(\frac{c^2-4c+4}{12c^3+30c^2}\div\frac{c^2-4}{6c^4+15c^3}\)
Dividing by a fraction is the same as multiplying by its reciprocal.
\(\frac{c^2-4c+4}{12c^3+30c^2}*\frac{6c^4+15c^3}{c^2-4}\)
Now, let's factor the numerators and denominators completely and fully and see if any canceling can occur to simplify this.
| \(\frac{c^2-4c+4}{12c^3+30c^2}*\frac{6c^4+15c^3}{c^2-4}\) | Factor everything fully. |
| \(\frac{(c-2)^2}{6c^2(2c+5)}*\frac{3c^3(2c+5)}{(c+2)(c-2)}\) | I see a lot of canceling that will occur here, dont you? |
| \(\frac{c-2}{2}*\frac{c}{c+2}\) | Now, combine. |
| \(\frac{c(c-2)}{2(c+2)}\) | This answer corresponds to the third one listed in the multiple guess. |
