Answered on the original post :)
Hi Guest!
\(xy=x+y \\ xz=2(x+z) \\ yz=3(y+z)\)
\(x=\frac{y}{y-1}, \text{ From the first equation}\)
Now, substituting in the second:
\(\frac{y}{y-1}z=2(\frac{y}{y-1}+z)\)
Then, solving for z:
\(yz=2y+2(y-1)z \\ \implies z(y-2(y-1))=2y \\ z=\frac{2y}{2-y}\)
Now, substituting this in the third equation:
\(y(\frac{2y}{2-y})=3(y+\frac{2y}{2-y}) \\ \implies 2y^2=3(y)(2-y)+3(2y) \\ \implies 2y^2=6y-3y^2+6y \\ \implies 5y^2-12y=0 \implies y(5y-12)=0 \implies y_1=0 \text{ or } y_2=\frac{12}{5}\)
Case 1: \(y=0 \implies z=0,x=0\) (Rejected).
Case 2: \(y=\frac{12}{5} \implies z=-12 \implies x=\frac{12}{7}\)
We reject case 1, as it will not satisfy the initial equations given. That is, the denominators will be 0.
Therefore, \((x,y,z):=(\frac{12}{7},\frac{12}{5},-12)\)
Thus, \(z=-12\)
I hope this helps :)!
To simplify a fraction, multiply/divide the numerator and denominator by the same number.
2*3*14=84
8/36 simplifies to 2/9
(0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5)>>Total = 36
The probability is: 8 / 36 ==4 / 9
A.85 1414 cubic feet
17 yellow highlighters
17 + 4 ==21 pink highlighters
21 + 9 ==30 blue highlighters.
17 + 21 + 30 ==68 - total number of highlighters in Braden's desk.
3 gallons x 2 ==6 gallons of insecticide per acre.
6 x 14 ==84 gallons of insecticide will be needed to spray 14 acres.
1,320 / 24 ==55 full boxes of zucchinis.
55 x $2.77 ==$152.35
$152.35 / 2 ==$76.18 - what Mr. Brown will get.
The average round trip speed is always 57 MPH or this journey
Total round trip distance is 156 miles.
The Bus went from Bigtown to Megacity and is returning to Bigtown and has completed 213 of the entire journey.
i.e, the bus has covered a distance equal to.
2/3 * 156 = 104 miles
Time required for completing 104 miles is:
Distance/Velocity = 109/57 = 1.825 hours
= 1.825 * 60 minutes
= 109.5 minutes
Again the Bus waits exactly 25 min at the Megacity. So, total time as elapsed since the Bus left Bigtown on the way to the Megacity is:
109.5 + 25 = 134.5 minutes
= 135 minutes
(Rounding off to the nearest minute)
Erin's Donuts: $0.40 per donut, $1.50 off for a dozen
1 dozen = 12 units
so, a dozen cost = 0.4 * 12 - 15 = 4.8 - 1.5 = $3.3
16 - 12 = 4 donuts
so, Erin cost = 0.4 * 4 + 3.3 = 1.6 + 3.3 = $4.9
Anne's Donuts: $0.5 per donut, $2.10 off for a dozen
so, a dozen cost = 0.5 * 12 - 2.10 = 6 - 2.10 = $3.9
so, Anne cost = 0.5 * 4 + 3.9 = 2 + 3.9 = $5.9
because 4.9 < 5.9
so, Erin Donuts cost the least.
dividing the numerator and the denominator by the largest number that can divide into both numbers exactly. That is the most basic way to simplify basic fractions. For example expressions/exponents would not completely work. Ex: 10/5 = 2/1 or just 2
10/35 = 2/7 If you would like to simplify exponents and expressions there are other ways to approach that
Thank you so much BuilderBoy!
420 might be incorrect
From the first equation and second equations, we have \(p = 2q\) and \(q = 2r\), meaning \(p = 2 \times 2r = 4r\)
Substituting this into the third equation gives us \(r = 1\). Plugging this value into the second equation gives us \(q - 2 = 0\), meaning \(q = 2\)
Finally, substituting \(q = 2\) into the first equation \(p - 4 = 0 \), meaning \(p = 4\)
So, the solution is \(\color{brown}\boxed{(4,2,1)}\)
There are \(8! \div 8 = 7! \) ways to put the mages, but we have to divide by \(2! \times 3! = 12\) to get \({7! \div 12} = \color{brown}\boxed{420}\)
That address is not valid
7 apples cost $4.00, 9 oranges cost $11.00. James bought a total of 1000 apples and oranges for $999.00. How many apples and how many oranges did he buy?
A==Apples, 1000 - A==Oranges
$4 / 7 ==Price of 1 apple
$11 / 9 ==Price of 1 orange.
4/7A + 11/9 * [1000 - A] ==999.00, solve for A
A==343 apples that James bought
1000 - 343 ==657 oranges that James bought.
Check: [343 / 7 * $4] + [657/9 * $11]==$196.00 + $803.00==$999.00
81 books are remaining.
7:09 - 5:41 = (7 - 5) : (9 - 41)
= 2 : (-32)
= 1 : (60 - 32)
= 1 : 28
1 hour 28 minutes
x - 1/4x - 7 = 23
x - 1/4x = 23 + 7
3/4x = 30
x = 40
Thanks!
1/J + 1/S ==1/12
But J == S
1/J + 1/J ==1/12, solve for J
J == 24 hours - time it takes Jack working alone to write the lab report
S == 24 hours - ...................Serena..........................................................
Kay ==24 hours as well:
Therefore: Jack, Serena and Kay would finish the report in:
3 / 24 ==8 hours
