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Ivahew [28]
3 years ago
10

What are these fractions in simplest form?

Mathematics
1 answer:
Anton [14]3 years ago
8 0
\frac{16y^{3}}{20y^{4}} =  \frac{4}{5y}
\frac{6xy}{16y} =  \frac{3x}{8}
\frac{abc}{10abc} =  \frac{1}{10}
\frac{mn^{2}}{pm^{5}n} =  \frac{n}{pm^{4}}
\frac{12h^{3}k}{16h^{2}k^{2}} =  \frac{3h}{4k}
\frac{8x}{10y} =  \frac{4x}{5y}
\frac{24n^{2}}{28n} =  \frac{6n}{7}
\frac{30hxy}{54kxy}  =  \frac{5h}{9k}
\frac{5jh}{15jh^{3}} =  \frac{1}{3h^{2}}
\frac{20s^{2}t^{3}}{16st^{5}} =  \frac{5s}{4t^{2}}
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(6x-5y+4)dy+(y-2x-1)dx=0​
Len [333]

(6<em>x</em> - 5<em>y</em> + 4) d<em>y</em> + (<em>y</em> - 2<em>x</em> - 1) d<em>x</em> = 0

(6<em>x</em> - 5<em>y</em> + 4) d<em>y</em> = (2<em>x</em> - <em>y</em> + 1) d<em>x</em>

d<em>y</em>/d<em>x</em> = (2<em>x</em> - <em>y</em> + 1) / (6<em>x</em> - 5<em>y</em> + 4)

Let <em>X</em> = <em>x</em> - <em>a</em> and <em>Y</em> = <em>y</em> - <em>b</em>. We want to find constants <em>a</em> and <em>b</em> such that

d<em>Y</em>/d<em>X</em> = (a rational function)

where the numerator and denominator on the right side are free of constant terms. Substituting <em>x</em> and <em>y</em> in the equation, we have

d<em>Y</em>/d<em>X</em> = (2 (<em>X</em> + <em>a</em>) - (<em>Y</em> + <em>b</em>) + 1) / (6 (<em>X</em> + <em>a</em>) - 5 (<em>Y</em> + <em>b</em>) + 4)

d<em>Y</em>/d<em>X</em> = (2<em>X</em> - <em>Y</em> + 2<em>a</em> - <em>b</em> + 1) / (6<em>X</em> - 5<em>Y</em> + 6<em>a</em> - 5<em>b</em> + 4)

Then we solve for <em>a</em> and <em>b</em> in the system,

2<em>a</em> - <em>b</em> + 1 = 0

6<em>a</em> - 5<em>b</em> + 4 = 0

==>   <em>a</em> = -1/4 and <em>b</em> = 1/2

With these constants, the equation reduces to

d<em>Y</em>/d<em>X</em> = (2<em>X</em> - <em>Y</em>) / (6<em>X</em> - 5<em>Y</em>)

Now substitute <em>Y</em> = <em>VX</em> and d<em>Y</em>/d<em>X</em> = <em>X</em> d<em>V</em>/d<em>X</em> + <em>V</em> :

<em>X</em> d<em>V</em>/d<em>X</em> + <em>V</em> = (2<em>X</em> - <em>VX</em>) / (6<em>X</em> - 5<em>VX</em>)

The equation becomes separable after some simplification:

<em>X</em> d<em>V</em>/d<em>X</em> + <em>V</em> = (2 - <em>V</em>) / (6 - 5<em>V</em>)

<em>X</em> d<em>V</em>/d<em>X</em> = (2 - <em>V</em>) / (6 - 5<em>V</em>) - <em>V</em>

<em>X</em> d<em>V</em>/d<em>X</em> = (2 - <em>V</em> - (6 - 5<em>V</em>)) / (6 - 5<em>V</em>)

<em>X</em> d<em>V</em>/d<em>X</em> = (4<em>V</em> - 4) / (6 - 5<em>V</em>)

- (5<em>V</em> - 6) / (4<em>V</em> - 4) d<em>V</em> = 1/<em>X</em> d<em>X</em>

Integrate both sides:

-5/4 <em>V</em> + 1/4 ln|4<em>V</em> - 4| = ln|<em>X</em>| + <em>C</em>

Extract a constant from the logarithm on the left:

-5/4 <em>V</em> + 1/4 (ln(4) + ln|<em>V</em> - 1|) = ln|<em>X</em>| + <em>C</em>

-5/4 <em>V</em> + 1/4 ln|<em>V</em> - 1| = ln|<em>X</em>| + <em>C</em>

-5<em>V</em> + ln|<em>V</em> - 1| = 4 ln|<em>X</em>| + <em>C</em>

Get this back in terms of <em>Y</em> :

-5<em>Y/X</em> + ln|<em>Y/X</em> - 1| = 4 ln|<em>X</em>| + <em>C</em>

Now get the solution in terms of <em>y</em> and <em>x</em> :

-5 (<em>y</em> - 1/2)/(<em>x</em> + 1/4) + ln|(<em>y</em> - 1/2)/(<em>x</em> + 1/4) - 1| = 4 ln|<em>x</em> + 1/4| + <em>C</em>

<em />

With some manipulation of constants and logarithms, and a bit of algebra, we can rewrite this solution as

-5 (4<em>y</em> - 2)/(4<em>x</em> + 1) + ln|(4<em>y</em> - 4<em>x</em> - 3)/(4<em>x</em> + 1)| = 4 ln|<em>x</em> + 1/4| + 4 ln(4) + <em>C</em>

-5 (4<em>y</em> - 2)/(4<em>x</em> + 1) + ln|(4<em>y</em> - 4<em>x</em> - 3)/(4<em>x</em> + 1)| = 4 ln|4<em>x</em> + 1| + <em>C</em>

-5 (4<em>y</em> - 2)/(4<em>x</em> + 1) + ln|4<em>y</em> - 4<em>x</em> - 3| - ln|4<em>x</em> + 1| = 4 ln|4<em>x</em> + 1| + <em>C</em>

-5 (4<em>y</em> - 2)/(4<em>x</em> + 1) + ln|4<em>y</em> - 4<em>x</em> - 3| = 5 ln|4<em>x</em> + 1| + <em>C</em>

8 0
3 years ago
HELP ME PLEASEEEEEEEEEEEE
nasty-shy [4]
The answer is 0.1

17/10 - 8/5 = 0.1
7 0
3 years ago
3.1(p - 2.7) simplified:
Mashcka [7]

3.1(p - 2.7) simplified:= 3.1p-8.37

7 0
3 years ago
Read 2 more answers
For which values of x will the rational expression below be undefined? Check
ololo11 [35]

Answer:

-2

Step-by-step explanation:

For the given expression to be undefined, this means that the denominator must be zero.

From the given expression, the denominator is x + 2

Equating this to zero;

x+2 = 0

Subtract 2 from both sides

x+2-2 = 0 -2

x = -2

Hence the value of x that makes it undefined is -2

3 0
3 years ago
Can y’all help plz thx
Alexus [3.1K]

Answer:

C is the answer

3 0
3 years ago
Read 2 more answers
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