Question

What is the solution of the system of equations? 3x+2y+z=7 5x+5y+4z=3 3x+2y+3z=1

Answer 1

Solution :

{x,y,z} = {4,-1,-3}

System of Linear Equations entered :

[1] 3x + 2y + z = 7

[2] 5x + 5y + 4z = 3

[3] 3x + 2y + 3z = 1

Solve by Substitution :

// Solve equation [1] for the variable z

[1] z = -3x - 2y + 7

// Plug this in for variable z in equation [2]

[2] 5x + 5y + 4•(-3x-2y+7) = 3

[2] -7x - 3y = -25

// Plug this in for variable z in equation [3]

[3] 3x + 2y + 3•(-3x-2y+7) = 1

[3] -6x - 4y = -20

// Solve equation [3] for the variable y

[3] 4y = -6x + 20

[3] y = -3x/2 + 5

// Plug this in for variable y in equation [2]

[2] -7x - 3•(-3x/2+5) = -25

[2] -5x/2 = -10

[2] -5x = -20

// Solve equation [2] for the variable x

[2] 5x = 20

[2] x = 4

// By now we know this much :

x = 4

y = -3x/2+5

z = -3x-2y+7

// Use the x value to solve for y

y = -(3/2)(4)+5 = -1

// Use the x and y values to solve for z

z = -3(4)-2(-1)+7 = -3

Solution :

{x,y,z} = {4,-1,-3}

Hope this has been explanatory enough

Answer 2

The answer is x = 4
y = -1
z = -3

3x+2y+z=7
5x+5y+4z=3
3x+2y+3z=1
________
Let's take the first and the third equation, multiply the first one by (-1) and sum them up:
3x+2y+z=7
3x+2y+3z=1
____
-3x-2y-z=-7
3x+2y+3z=1
____
2z = -6
z = -6/2
z = -3


Substitute z in all equations:
3x+2y+z=7
5x+5y+4z=3
3x+2y+3z=1
______
3x + 2y - 3 = 7
5x + 5y - 12 = 3
3x + 2y - 9 = 1
______
3x + 2y = 10
5x + 5y = 15
3x + 2y = 10

The first and the third equations are the same, so one can be excluded:
3x + 2y = 10
5x + 5y = 15

Multiply the first equation by -5 and the second equation by 2 and sum them up:
-15x - 10y = -50
10x + 10y = 30
_________
-5x = -20
x = -20/-5
x = 4

Substitute x:
3x + 2y = 10
12 + 2y = 10
2y = 10 - 12
2y = -2
y = -2/2
y = -1