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Aleonysh [2.5K]

2 years ago

13

Solve the set of equations x-y=4 2x+y=9

1 answer:

pochemuha2 years ago

5 0

X - y = 4 (1)
2x + y = 9 (2)
We need to solve for x and y.

Let's find the algebraic value of x using the equation (1):
x-y = 4
add y in both sides
x-y+y = 4+y
x = 4+y

Now let's using this value instead of x in the equation (2):
2x + y = 9
2(4+y) + y = 9
8 + 2y + y = 9
3y + 8 = 9
Subtract 8 from both sides
3y + 8 - 8 = 9 - 8
3y = 1
divide both sides by 3
(3y)/3 = 1/3
y=1/3

Let's use the value of y to find the value x using the equation (1)
x - y = 4
x = y+4
x = 1/3 + 4
x = 13/3

So x = 13/3 and y = 1/3

Let's recheck the answers (very important):
x - y = 13/3 - 1/3 = 12/3 = 4 (correct)
2x + y = 2*(13/3) + 1/3 = 26/3 + 1/3 = 27/3 = 9 (correct)
The answer has been approved.

Hope this helps! :D

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Find the general solution to 1/x dy/dx - 2y/x^2 = x cos x, y(pi) = pi^2

Finger [1]

Answer:

$\frac{y}{x^2}=\sin x+\pi$

Step-by-step explanation:

Consider linear differential equation $\frac{\mathrm{d} y}{\mathrm{d} x}+yp(x)=q(x)$

It's solution is of form $y\,I.F=\int I.F\,q(x)\,dx$ where I.F is integrating factor given by $I.F=e^{\int p(x)\,dx}$ .

Given: $\frac{1}{x}\frac{\mathrm{d} y}{\mathrm{d} x}-\frac{2y}{x^2}=x\cos x$

We can write this equation as $\frac{\mathrm{d} y}{\mathrm{d} x}-\frac{2y}{x}=x^2\cos x$

On comparing this equation with $\frac{\mathrm{d} y}{\mathrm{d} x}+yp(x)=q(x)$ , we get $p(x)=\frac{-2}{x}\,\,,\,\,q(x)=x^2\cos x$

I.F = $e^{\int p(x)\,dx}=e^{\int \frac{-2}{x}\,dx}=e^{-2\ln x}=e^{\ln x^{-2}}=\frac{1}{x^2}$ { formula used: $\ln a^b=b\ln a$ }

we get solution as follows:

$\frac{y}{x^2}=\int \frac{1}{x^2}x^2\cos x\,dx\\\frac{y}{x^2}=\int \cos x\,dx\\\\\frac{y}{x^2}=\sin x+C$

{ formula used: $\int \cos x\,dx=\sin x$ }

Applying condition: $y(\pi)=\pi^2$

$\frac{y}{x^2}=\sin x+C\\\frac{\pi^2}{\pi}=\sin\pi+C\\\pi=C$

So, we get solution as :

$\frac{y}{x^2}=\sin x+\pi$

4 0

3 years ago

A cylinder has a height of 16 yards. Its volume is 6,079.04 cubic yards. What is the radius

Katen [24]

Answer:

The radius of the cylinder is: r = 11 yards

Step-by-step explanation:

Given

The height of cylinder = h = 16 yards

Volume of cylinder = V = 6079.04 cubic yards

To determine

The radius of cylinder = r = ?

Using the formula involving volume of the cyliner

V = πr²h

substituting h = 16, V = 6079.04 and π = 3.14

6079.04 = 3.14 (r²) 16

r² = [6079.04] / [(3.14) (16)]

r² = 6079.04 / 50.24

r² = 121

$r=\sqrt{121},\:r=-\sqrt{121}$

$r=11,\:r=-11$

As radius can not be negative.

Therefore, the radius of the cylinder is: r = 11 yards

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