Solve the system of equations

Question

5/4x - 4/3y + 5/4z = 6

5/16x + 5/4y - 4/16z = -4

3/4x - y +3/4z = 6

Answer 1

Multiply the third equation by (-5/3) and add to the first equation to cancel out x and z:

 

5/4x -4/3y +5/4z = 6

-5/4x +5/3y -5/4z = -10

 

Leaving you with 1/3y = -4  => y = -12

 

Plug -12 in for y in either the first or third equation and solve for x in terms of z:

 

5/4x -4/3(-12) +5/4z = 6

5/4x + 5/4z = -10

x = -z - 8

 

Plug (-z - 8) in for x and -12 in for y in the second equation and solve for z:

 

5/16(-z - 8) +5/4(-12) -4/16z = -4

-9/16z - 5/2 - 15 -4/16z = -4

-9/16z = 27/2

z = -24

 

x= 16

y= -12

z= -24

Answer 2

5/4x - 4/3y + 5/4z = 6
5/16x + 5/4y - 4/16z = -4
3/4x - y +3/4z = 6

Convert to Row Notation:
Each equation becomes a row.
Just like equations, rows can
be multiplied by a number,
added to each other, and moved.
The goal is to get just one
number in the first 3 of
each row that is not 0.

R1: {5/4,–4/3,5/4,6}
R2: {5/16,5/4,–4/16,–4}
R3: {3/4,–1,3/4,6}

R1 > R1*12
R2 > R2*16
R3 > R3*4

R1: {15,–16,15,72}
R2: {5,20,–4,–64}
R3: {3,–4,3,24}

R2 > -3*R2
R3 > -5*R3

R1: {15,–16,15,72}
R2: {-15,-60,12,192}
R3: {-15,20,-15,-120}

R2 > R2 + R1
R3 > R3 + R1

R1: {15,–16,15,72}
R2: {0,-76,27,264}
R3: {0,4,0,-48}

R3 > 19*R3

R1: {15,–16,15,72}
R2: {0,-76,27,264}
R3: {0,76,0,-912}

R3 > R3 + R2

R1: {15,–16,15,72}
R2: {0,-76,27,264}
R3: {0,0,27,-648}

R2 > R2 - R3

R1: {15,–16,15,72}
R2: {0,-76,0,912}
R3: {0,0,27,-648}

R2 > R2/(-76)
R3 > R3/27

R1: {15,–16,15,72}
R2: {0,1,0,-12}
R3: {0,0,1,-24}

R3 > –15*R3

R1: {15,–16,15,72}
R2: {0,1,0,-12}
R3: {0,0,–15,360}

R1 > R1 + R3

R1: {15,–16,0,432}
R2: {0,1,0,-12}
R3: {0,0,–15,360}

R2 > 16*R2
R3 > R3/(–15)

R1: {15,–16,0,432}
R2: {0,16,0,-192}
R3: {0,0,1,-24}

R1 > R1 + R2
R2 > R2/16

R1: {15,0,0,240}
R2: {0,1,0,-12}
R3: {0,0,1,-24}

R1 > R1/15

R1: {1,0,0,16}
R2: {0,1,0,-12}
R3: {0,0,1,-24}

(16,-12,-24) is solution.

check:

(5/4)(16) - (4/3)(-12) + (5/4)(-24) =? 6
(5/16)(16) + (5/4)(-12) - (4/16)(-24) =? -4
(3/4)(16) - (-12) + (3/4)(-24) =? 6

20 + 16 – 30 = 6 √
5 - 15 + 6 = -4 √
12 + 12 - 18 = 6 √