Find the values of x, y, and z that satisfy the system of equations:
5x – 3y + 7z = 22; 3x – 5y – 2z = –46; 2x – 2y + 5z = 24.

Question

Find the values of x, y, and z that satisfy the system of equations:   
5x – 3y + 7z = 22; 3x – 5y – 2z = –46; 2x – 2y + 5z = 24.

Accepted Answer

We are given the system of equations:

1) 5x – 3y + 7z = 22
2) 3x – 5y – 2z = –46
3) 2x – 2y + 5z = 24

Step 1: Use equations (1) and (2) to eliminate one variable.

Multiply equation (1) by 3 and equation (2) by 5 to eliminate x:

(1)*3: 15x – 9y + 21z = 66
(2)*5: 15x – 25y – 10z = –230

Subtract (2)*5 from (1)*3:
(15x – 9y + 21z) – (15x – 25y – 10z) = 66 – (–230)
=> 0x + 16y + 31z = 296

So, 16y + 31z = 296  ---- (4)

Step 2: Use equations (1) and (3) to eliminate x.

Multiply equation (1) by 2 and equation (3) by 5:

(1)*2: 10x – 6y + 14z = 44
(3)*5: 10x – 10y + 25z = 120

Subtract (1)*2 from (3)*5:
(10x – 10y + 25z) – (10x – 6y + 14z) = 120 – 44
=> 0x – 4y + 11z = 76

So, -4y + 11z = 76  ---- (5)

Step 3: Solve equations (4) and (5):

From (5): -4y + 11z = 76
=> 4y = 11z – 76
=> y = (11z – 76)/4

Substitute y in (4):
16y + 31z = 296
16 * ((11z – 76)/4) + 31z = 296
=> 4 * (11z – 76) + 31z = 296
=> 44z – 304 + 31z = 296
=> 75z – 304 = 296
=> 75z = 600
=> z = 600 / 75 = 8

Step 4: Find y:

y = (11*8 – 76)/4 = (88 – 76)/4 = 12/4 = 3

Step 5: Find x using equation (1):

5x – 3y + 7z = 22
5x – 3*3 + 7*8 = 22
5x – 9 + 56 = 22
5x + 47 = 22
5x = 22 – 47 = –25
x = –25 / 5 = –5

Therefore, the solution is x = –5, y = 3, z = 8.

Answer

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