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Question

What is the solution to the following system of equations? (If using a matrix calculator, use the image icon below to upload your work)

x−3y+2z=11
-x+4y+3z=5
2x−2y−4z=2

Answer 1

To solve the system of equations, we can use the method of elimination.

First, let's eliminate the x variable by adding the first equation to the second equation:

(x - 3y + 2z) + (-x + 4y + 3z) = 11 + 5

Simplifying this equation, we get:

y + 5z = 16 ...........(1)

Next, let's eliminate the x variable by adding twice the first equation to the third equation:

2(x - 3y + 2z) + (2x - 2y - 4z) = 2 + 2

Simplifying this equation, we get:

-4y + 2z = 6 .............(2)

Now we have two equations in terms of y and z:

y + 5z = 16 ...........(1)
-4y + 2z = 6 .............(2)

To eliminate y, we can multiply the first equation by 4 and the second equation by 5:

4(y + 5z) = 4(16)
-5(-4y + 2z) = 5(6)

This gives us:

4y + 20z = 64 ............(3)
20y - 10z = 30 ...........(4)

Now we can add equations (3) and (4) to eliminate y:

(4y + 20z) + (20y - 10z) = 64 + 30

Simplifying this equation, we get:

24y + 10z = 94 ...........(5)

Next, let's multiply equation (1) by 24:

24(y + 5z) = 24(16)

This gives us:

24y + 120z = 384 ............(6)

Now we can subtract equation (6) from equation (5) to eliminate y:

(24y + 10z) - (24y + 120z) = 94 - 384

Simplifying this equation, we get:

-110z = -290

Dividing both sides by -110, we find:

z = 2.63636363636... ≈ 2.64

Substitute this value of z back into equation (1):

y + 5(2.64) = 16

Simplifying this equation, we get:

y + 13.2 = 16

Subtracting 13.2 from both sides, we have:

y = 2.8

Finally, substitute the values of y and z into one of the original equations (let's use the first equation):

x - 3(2.8) + 2(2.64) = 11

Simplifying this equation, we get:

x - 8.4 + 5.28 = 11

Combining like terms, we have:

x - 3.12 = 11

Adding 3.12 to both sides, we find:

x = 14.12

Therefore, the solution to the system of equations is:

x = 14.12
y = 2.8
z = 2.64