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Describe each system of planes. If possible, solve the

system.

a) 3x + 2y - z = -2

2x + y - 2z = 7

2x - 3y + 4z = -3

b) 3x - 4y + 2z = 1

6x - 8y + 4z = 10

15x - 20y + 10z = -3

c) 2x + y + 6z = 5

5x +

User Jmn
by
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1 Answer

2 votes

Answer:

Explanation:

a) The system of planes is represented by the following equations:

3x + 2y - z = -2

2x + y - 2z = 7

2x - 3y + 4z = -3

To solve this system, we can use various methods such as substitution, elimination, or matrix methods. Here, we'll use the elimination method to find the solution:

Step 1: Multiply the second equation by 2 and the third equation by 3 to create a cancellation:

3x + 2y - z = -2

4x + 2y - 4z = 14

6x - 9y + 12z = -9

Step 2: Subtract the first equation from the second equation and the first equation from the third equation:

4x + 2y - 4z - (3x + 2y - z) = 14 - (-2)

6x - 9y + 12z - (3x + 2y - z) = -9 - (-2)

Simplifying, we get:

x - 3z = 16 (Equation 4)

3x - 11y + 13z = -7 (Equation 5)

Step 3: Multiply Equation 4 by 3 and add it to Equation 5:

3(x - 3z) + (3x - 11y + 13z) = 16(3) - 7

Simplifying, we have:

6x - 2y + 4z = 37 (Equation 6)

Step 4: Now we have the following two equations:

6x - 2y + 4z = 37 (Equation 6)

2x + y - 2z = 7 (Equation 2)

Step 5: We can now solve this system of equations. One way is to multiply Equation 2 by 2 and add it to Equation 6:

2(2x + y - 2z) + (6x - 2y + 4z) = 2(7) + 37

Simplifying, we get:

10x = 51

Dividing both sides by 10, we find:

x = 5.1

Step 6: Substituting the value of x into Equation 2, we can solve for y:

2(5.1) + y - 2z = 7

Simplifying, we have:

y - 2z = -3.2 (Equation 7)

Step 7: Substituting the values of x and y into Equation 6, we can solve for z:

6(5.1) - 2y + 4z = 37

Simplifying, we get:

4z = 1.8

Dividing both sides by 4, we find:

z = 0.45

Therefore, the solution to the system of planes is x = 5.1, y = -3.2, and z = 0.45.

b) The system of planes is represented by the following equations:

3x - 4y + 2z = 1

6x - 8y + 4z = 10

15x - 20y + 10z = -3

To solve this system, we'll again use the elimination method:

Step 1: Multiply the first equation by 2 and the second equation by 5 to create a cancellation:

6x - 8y + 4z = 2

30x -

I hope this is right!! :)

User RmIu
by
8.4k points