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1 year ago

Find the GCF of 18xy and 32xyz

Mathematics

1 answer:

Tanzania [10]1 year ago

4 0

<h3>Answer:</h3>

2xy

<h3>Step-by-step explanation:</h3>

GCF stands for the greatest common factor.

Finding All Factors

To find the GCF you need to find the factors of each term. First, find the factors of the coefficients.

The factors of 18 include: 1, 2, 3, 6, 9 and 18

The factors of 32 include: 1, 2, 4, 8, 16, and 32

Next, lay out the variables that are factors.

The variables of 18xy are x and y.
The variables of 32xyz are x, y, and z.

Remember that variables count as factors

Greatest Common Factor

Now that we have all of the factors we can find the GCF. The GCF will be the largest number and the most variables that are a factor of both terms.

In this case, the largest factor of 18 and 32 is 2. Since both x and y are factors of both terms, the GCF is 2xy.

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A first order differential equation $y'=f(x,y)$ is called a separable equation if the function $f(x,y)$ can be factored into the product of two functions of $x$ and $y$ :

$f(x,y)=p(x)h(y)$

where $p(x)$ and $h(y)$ are continuous functions.

We have the following differential equation

$y'=(1-5x)y^2, \quad y(0)=-12$

In the given case $p(x)=1-5x$ and $h(y)=y^2$ .

We divide the equation by $h(y)$ and move $dx$ to the right side:

$\frac{1}{y^2}dy\:=(1-5x)dx$

Next, integrate both sides:

$\int \frac{1}{y^2}dy\:=\int(1-5x)dx\\\\-\frac{1}{y}=x-\frac{5x^2}{2}+C$

Now, we solve for $y$

$-\frac{1}{y}=x-\frac{5x^2}{2}+C\\-\frac{1}{y}\cdot \:2y=x\cdot \:2y-\frac{5x^2}{2}\cdot \:2y+C\cdot \:2y\\-2=2yx-5yx^2+2Cy\\y\left(2x-5x^2+2C\right)=-2\\\\y=-\frac{2}{2x-5x^2+2C}$

We use the initial condition $y(0)=-12$ to find the value of C.

$-12=-\frac{2}{2\left(0\right)-5\left(0\right)^2+2C}\\-12=-\frac{1}{c}\\c=\frac{1}{12}$

Therefore,

$y=-\frac{2}{2x-5x^2+2(\frac{1}{12})}\\y=-\frac{12}{12x-30x^2+1}$

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