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3 years ago

Solve the following system of equations: 2x + 4y − 3z = −7 3x + y + 4z = −12 x + 3y + 4z = 4

Mathematics

2 answers:

lukranit [14]3 years ago

4 0

2x + 4y − 3z = −7 ---> first equation

3x + y + 4z = −12 ---> second equation

x + 3y + 4z = 4 ---> third equation

We try to eliminate z and make two equations

Multiply first equation by 4 and second equation by 3. then add both equations

8x + 16y -12z = -28

9x + 3y +12z = -36

----------------------------

17x + 19y = -64 ---------------> equation 4

multiply third equation by -1 and add second and third equation

3x + y + 4z = −12

-x - 3y - 4z = -4

-------------------------------------

2x -2y = -16 ---------------> equation 5

Multiply equation 4 by 2 and equation by 19

34x + 38y = -128

38x - 38y = -304

------------------------------------

72x = -432

Divide by 72 on both sides

x = -6

use equation 2x -2y = -16 ---------------> equation 5

2(-6) - 2y = -16

-12 -2y = -16

-2y = -4

So y = 2

Now plug in 2 for y and -6 for x

Using any one of the three equations

x + 3y + 4z = 4

-6 + 3(2) +4z = 4

4z = 4 (divide by 4 on both sides)

So z=1

So x= -6, y = 2 and z= 1

Sauron [17]3 years ago

4 0

Answer:

(-6,2,1)

Step-by-step explanation:

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3 years ago

Solve 3x+2y−z=2 2y+z=7 −2x−4z=−2 Enter your answer, in the form (x,y,z), in the boxes in simplest terms. x= y= z=

Amanda [17]

Answer:

(-1, 3, 1)

x=-1, y=3, z=1.

Step-by-step explanation:

So we have the three equations:

$3x+2y-z=2\\2y+z=7\\-2x-4z=-2$

And we want to find the value of each variable.

To do so, look at the equations and try to think about what we can do to isolate one of the variables by itself.

We can see that both the second and equation has a variable and then the variable z.

In other words, we can solve for y in the second equation and x in the third equation and then substitute them in the first equation to isolate the variable z and then solve for z. With that, we can then easily arrive at our solution.

Therefore, solve for y in the second equation:

$2y+z=7$

Subtract z from both sides. The zs on the left side cancels:

$(2y+z)-z=(7)-z\\2y=7-z$

Now, divide both sides by 2. The 2s on the left cancel and we've isolated the y variable:

$\frac{(2y)}{2} = \frac{(7-z)}{2}\\y=\frac{(7-z)}{2}$

Now, do the same for the third equation. Isolate the x variable:

$-2x-4z=-2$

First, divide everything by -2 to make things simpler:

$\frac{(-2x-4z)}{-2}=\frac{(-2)}{-2}\\ x+2z=1$

Now, subtract 2z from both sides to isolate the x variable:

$(x+2z)-2z=1-2z\\x=1-2z$

We have isolated the x and y variables. They are:

$y=\frac{(7-z)}{2}\\x=1-2z$

Now, we can substitute them back into the first equation. Therefore:

$3x+2y-z=2\\3(1-2z)+2(\frac{7-z}{2})-z=2$

Distribute the first and second terms. The second term cancels out:

$3(1)-3(2z)+(7-z)-z=2\\3-6z+7-z-z=2$

Combine like terms:

$-6z-z-z+3+7=2\\-8z+10=2\\$

Subtract 10 from both sides:

$(-8z+10)-10=2-10\\-8z=-8$

Divide both sides by -8:

$z=1$

Now that we've determined that z=1, plug it back into the isolated second and third equations:

Second equation:

$y=\frac{(7-z)}{2}\\y=(7-1)/2\\y=(6)/2=3$

Third equation:

$x=1-2z\\x=1-2(1)\\x=1-2=-1$

Therefore, our answer is (-1, 3, 1)

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The displacement of a 500 g mass, undergoing simple harmonic motion, is defined by the function :

Valentin [98]

To solve this problem, we have to write out the simple harmonic motion equation and then use the respective formula to solve. The maximum kinetic energy, maximum potential energy and the maximum mechanical energy is equal to 7.56J.

<h3>Simple Harmonic Motion</h3>

The given equation of the simple harmonic motion is

$x = 3.5sin(\frac{\pi}{2t} + \frac{5\pi}{4})\\$

Data;

ω = π/2
k = 1.254N/m

Solving this

$\frac{dx}{dt} = \theta = -3.5 * \frac{\pi}{2} cos (\frac{\pi }{2}t + \frac{5\pi}{4})\\$

Let's calculate the maximum velocity.

$V_m = \frac{3.5\pi}{2}$

This is only possible when cos θ = -1

The maximum kinetic energy is

$K_m = \frac{1}{2}mv_m^2 = \frac{1}{2}* \frac{500}{1000} * (\frac{7\pi}{4} )^2\\K_m = 7.56J$

The maximum kinetic energy is 7.56J

The maximum potential energy can be calculated as

$\omega^2 = \frac{k}{m} \\k = \omega^2 m \\k = \frac{\pi^2}{4} * \frac{500}{1000}\\ k= 1.254 N/m$

Using the value of spring constant, we can find the maximum potential energy

$P.E = \frac{1}{2}kx^2\\P.E = \frac{1}{2} * 1.234 * 3.5^2\\P.E = 7.56J$

The maximum potential energy is 7.56J

The maximum mechanical energy is equal to the sum of maximum potential energy and the maximum kinetic energy.

$ME = P.E + K.E\\ME = 7.56J$

The reason ME is equal to 7.56J is because when x is at maximum, v = 0 and makes kinetic energy equal to zero.

ME = 7.56J

From the calculations above, the maximum kinetic energy, maximum potential energy and the maximum mechanical energy is equal to 7.56J.

Learn more on simple harmonic motion here;

brainly.com/question/15556430

#SPJ1

3 0

2 years ago