ordered triple system of equations stumped?

Question

A) 3x+2y+z = -21
B) 2x-5y-z =7
C) 4x+y+5z= -43
The correct answer via TI-84+ is {-4, -2, -5}

But I am determined to figure this out by hand but so far the closest answer I have is
{4 1/13, 2 1/2, 12 3/4}

Steps I took:
1.) Added A and B
A) 3x +2y +z = -21
B) 2x  -5y  -z =   7
 to get:
    5x   -3y      = -14

2) Multiplied B by 5 then Added B and C
[B) 5(2x-5y-z =7)]
C) 4x       +y +5z = -43
B) 10x  -25y -5z =  35
 to get
    14x   -24y      =   -8

3) Took step 1's answer and multiplied by -8 and added to step 2's answer
[Step 1) [-8(5x     -3y      = -14)]
Step 2)  14x     -24y      =    -8
Step 1) -40x    +24y      = 114
 to get
-26x=106
x=-106/26 or 4 1/13 (previous thought it was 3 5/7 so y and z by my calcs are wrong (2 1/2, 12 3/4)

Obviously, I am messing up somewhere... I'd appreciate the help, thanks

None · Answer

A) 3x + 2y + z = -21
B) 2x - 5y - z =7
C) 4x + y+ 5z= -43

Add A and B to get 5x - 3y = -14........D)

Add 5(B) to C to get 14x - 24y = -8 and simplify to 7x - 12y = -4.......E)

Subtract E from 4(D) to get 13x = -52, giving x = -4 from which y = -2, z = -5 

You went wrong in your step 3)

Captain Matticus, LandPiratesInc · Answer

3x + 2y + z = -21
2x - 5y - z = 7
4x + y + 5z = -43


3x + 2y + z + 2x - 5y - z = -21 + 7
5x - 3y = -14
5x = 3y - 14
x = (1/5) * (3y - 14)


3x + 2y + z = -21
4x + y + 5z = -43

5 * (3x + 2y + z) = 5 * (-21)
15x + 10y + 5z = -105

15x + 10y + 5z - (4x + y + 5z) = -105 - (-43)
15x - 4x + 10y - y + 5z - 5z = -62
11x + 9y = -62
11x = -9y - 62
x = (-1/11) * (9y + 62)


x = (1/5) * (3y - 14)
x = (-1/11) * (9y + 62)


(3y - 14) / 5  =  -(9y + 62) / 11
11 * (3y - 14) = -5 * (9y + 62)
33y - 154 = -45y - 310
33y + 45y = 154 - 310
78y = -156
y = -156/78
y = -2

x = (1/5) * (3y - 14)
x = (1/5) * (-6 - 14)
x = (1/5) * (-20)
x = -4


3x + 2y + z = -21
-12 - 4 + z = -21
z - 16 = -21
z = -5

(-4 , -2 , -5)

bane · Answer

turn X, Y, Z a single variable like n. x + y + z = 6 2x + 4y + 2z = 24 -x + 8y - 3z = 36 n + n + n = 6 2n + 4n + 2n = 24 -n + 8n - 3n = 36 3n = 6 8n = 24 4n = 36 n = 2 n = 3 n = 9

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ordered triple system of equations stumped?

3 Answers