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Agata [3.3K]
3 years ago
13

Solve the following system of equations by linear combination:

Mathematics
1 answer:
Sholpan [36]3 years ago
3 0
3d - e =7
d  + e = 5
-------------
4d  /  = 12
d=12:4
d=3

3+e=5
e=5-3
e=2
the solution is (3,2)
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An environment engineer measures the amount ( by weight) of particulate pollution in air samples ( of a certain volume ) collect
Serggg [28]

Answer:

k = 1

P(x > 3y) = \frac{2}{3}

Step-by-step explanation:

Given

f \left(x,y \right) = \left{ \begin{array} { l l } { k , } & { 0 \leq x} \leq 2,0 \leq y \leq 1,2 y  \leq x }  & { \text 0, { elsewhere. } } \end{array} \right.

Solving (a):

Find k

To solve for k, we use the definition of joint probability function:

\int\limits^a_b \int\limits^a_b {f(x,y)} \, = 1

Where

{ 0 \leq x} \leq 2,0 \leq y \leq 1,2 y  \leq x }

Substitute values for the interval of x and y respectively

So, we have:

\int\limits^2_{0} \int\limits^{x/2}_{0} {k\ dy\ dx} \, = 1

Isolate k

k \int\limits^2_{0} \int\limits^{x/2}_{0} {dy\ dx} \, = 1

Integrate y, leave x:

k \int\limits^2_{0} y {dx} \, [0,x/2]= 1

Substitute 0 and x/2 for y

k \int\limits^2_{0} (x/2 - 0) {dx} \,= 1

k \int\limits^2_{0} \frac{x}{2} {dx} \,= 1

Integrate x

k * \frac{x^2}{2*2} [0,2]= 1

k * \frac{x^2}{4} [0,2]= 1

Substitute 0 and 2 for x

k *[ \frac{2^2}{4} - \frac{0^2}{4} ]= 1

k *[ \frac{4}{4} - \frac{0}{4} ]= 1

k *[ 1-0 ]= 1

k *[ 1]= 1

k = 1

Solving (b): P(x > 3y)

We have:

f(x,y) = k

Where k = 1

f(x,y) = 1

To find P(x > 3y), we use:

\int\limits^a_b \int\limits^a_b {f(x,y)}

So, we have:

P(x > 3y) = \int\limits^2_0 \int\limits^{y/3}_0 {f(x,y)} dxdy

P(x > 3y) = \int\limits^2_0 \int\limits^{y/3}_0 {1} dxdy

P(x > 3y) = \int\limits^2_0 \int\limits^{y/3}_0  dxdy

Integrate x leave y

P(x > 3y) = \int\limits^2_0  x [0,y/3]dy

Substitute 0 and y/3 for x

P(x > 3y) = \int\limits^2_0  [y/3 - 0]dy

P(x > 3y) = \int\limits^2_0  y/3\ dy

Integrate

P(x > 3y) = \frac{y^2}{2*3} [0,2]

P(x > 3y) = \frac{y^2}{6} [0,2]\\

Substitute 0 and 2 for y

P(x > 3y) = \frac{2^2}{6} -\frac{0^2}{6}

P(x > 3y) = \frac{4}{6} -\frac{0}{6}

P(x > 3y) = \frac{4}{6}

P(x > 3y) = \frac{2}{3}

8 0
3 years ago
There are 32 students in a class. Of the class, 3/8 of the students bring their own lunches. How many students bring their lunch
OverLord2011 [107]

Answer:

The number of students that bring their lunches is 12        

Step-by-step explanation:

Let

x -----> the number of students that bring their lunches

y -----> the total number of students in a class

we know that

The number of students that bring their lunches divided by  the total number of students in a class must be equal to 3/8

\frac{x}{y}=\frac{3}{8}

x=\frac{3}{8}y  -----> equation A

y=32 -----> equation B

substitute the value of y in equation A and solve for x

x=\frac{3}{8}(32)

x=12

therefore

The number of students that bring their lunches is 12

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3 years ago
Jared is using a 10ft rope
SVEN [57.7K]
Nice;-;

does jared need help carrying it
6 0
3 years ago
Two over twenty-two plus six over eleven
kirill [66]

\frac{2}{22}+\frac{6}{11}


Simplify \frac{2}{22} to \frac{1}{11}


\frac{1}{11}+\frac{6}{11}


Both are over 11, so, just sum.


\frac{1+6}{11}

\frac{7}{11}

4 0
3 years ago
Read 2 more answers
If the hypotenuse of a right triangle is 2 <img src="https://tex.z-dn.net/?f=%5Csqrt%7B3%7D" id="TexFormula1" title="\sqrt{3}" a
Svetradugi [14.3K]

Answer:

3

Step-by-step explanation:

a = the other leg:

a^{2} =(2\sqrt{3} )^{2} -(\sqrt{3}) ^{2} =(4)(3)-3=12-3

a^{2} =9

a=\sqrt{9}

a=3

Hope this helps

5 0
1 year ago
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