Question 152808
Hi, Hope I can help,
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solve the system:
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x+3y+5z=20
y-4z=(-16)
3x-2y+9z=36
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This is the way I usually solve these problems(pretty easy once you know how to do it) ( There is no fast way to solve these problems)
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First, solve for a letter in all the equations( since we already have a 2 system/variable equation for equation 2, we don't have to solve for any letter in equation 2)( Since {{{ y - 4z = (-16) }}} that means we will need to solve for "x" in our other equations)
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We will rewrite the 3 equations so it makes more sense( the second equation has no "x's" in the equation so it has "0x")
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x+3y+5z=20
0x + y - 4z =( -16)
3x-2y+9z=36 
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We will switch the equations around
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x+3y+5z=20
3x-2y+9z=36 
0x + y - 4z = (-16)
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We will solve "x" in our first two equations(can't solve "x" in our third equation, since it is 0x)
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First equation
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{{{ x+3y+5z=20 }}}
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We will move "3y" over to the right side
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{{{ x+3y+5z=20 }}} = {{{ x+3y - 3y +5z=20 -3y }}}
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{{{ x+3y - 3y +5z=20 -3y }}} = {{{ x + 5z= 20 - 3y }}} 
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We will move "5z" to the right side
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{{{ x + 5z= 20 - 3y }}} = {{{ x + 5z - 5z = 20 - 3y - 5z }}}
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{{{ x + 5z - 5z = 20 - 3y - 5z }}} = {{{ x = 20 - 3y - 5z }}}
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We will switch the letters around
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{{{ x = 20 - 3y - 5z }}} = {{{ x = (-3y) - 5z + 20 }}}
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This is our First Answer {{{ (-3y) - 5z + 20 }}}
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We will now solve "x" in our second equation
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{{{ 3x-2y+9z=36 }}}
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We will move (-2y) to the right side
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{{{ 3x-2y+9z=36 }}} = {{{ 3x-2y + 2y +9z=36 + 2y }}}
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{{{ 3x-2y + 2y +9z=36 + 2y }}} = {{{ 3x + 9z = 36 + 2y }}}
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We will move "9z" to the right side
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{{{ 3x + 9z = 36 + 2y }}} = {{{ 3x + 9z - 9z = 36 + 2y - 9z }}}
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{{{ 3x + 9z - 9z = 36 + 2y - 9z }}} = {{{ 3x = 36 + 2y - 9z }}}
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We will switch the letters around
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{{{ 3x = 36 + 2y - 9z }}} = {{{ 3x = 2y - 9z + 36 }}}
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We will now divide each side by "3"
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{{{ 3x = 2y - 9z + 36 }}} = {{{ 3x/3 = (2y - 9z + 36)/3 }}}
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{{{ 3x/3 = (2y - 9z + 36)/3 }}} = {{{ cross (3)x/cross(3) = (2y - 9z + 36)/3 }}}
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{{{ x = (2y - 9z + 36)/3 }}}
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Our second answer = {{{ (2y - 9z + 36)/3 }}}
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We can now solve to get a 2 system(variable) equation
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We will put our two answers together in an equation, since "x" = both of the answers, our answers equal each other
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First answer = {{{ (-3y) - 5z + 20 }}} 
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Second answer = {{{ (2y - 9z + 36)/3 }}}
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Our equation will equal {{{ (-3y) - 5z + 20 = (2y - 9z + 36)/3 }}} 
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{{{ (-3y) - 5z + 20 = (2y - 9z + 36)/3 }}} = {{{ ((-3y) - 5z + 20)/1 = (2y - 9z + 36)/3 }}}
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We will use cross multiplication to get rid of the fractions
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{{{ highlight((-3y) - 5z + 20)/1 = (2y - 9z + 36)/ highlight(3) }}}
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{{{ (-9y) - 15z + 60 = 2y - 9z + 36 }}}
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We will move (-9y) to the right side
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{{{ (-9y) + 9y - 15z + 60 = 2y + 9y - 9z + 36 }}}
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{{{ (-9y) + 9y - 15z + 60 = 2y + 9y - 9z + 36 }}} = {{{ (-15z) + 60 = 11y - 9z + 36 }}}
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We will move the (-15z) to the right side
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{{{ (-15z) + 60 = 11y - 9z + 36 }}} = {{{ (-15z) + 15z + 60 = 11y - 9z + 15z + 36 }}} 
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{{{ (-15z) + 15z + 60 = 11y - 9z + 15z + 36 }}} = {{{ 60 = 11y + 6z + 36 }}}
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We will move the "36" to the left side
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{{{ 60 = 11y + 6z + 36 }}} = {{{ 60 - 36 = 11y + 6z + 36 - 36 }}} 
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{{{ 60 - 36 = 11y + 6z + 36 - 36 }}} = {{{ 24 = 11y + 6z }}}
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We will rearrange, {{{ 24 = 11y + 6z }}} = {{{ 11y + 6z = 24 }}}
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Our second 2 system/variable equation = {{{ 11y + 6z = 24 }}}
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We will put our two 2 system/variable equations side by side 
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First equation = {{{ y-4z=(-16) }}}
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Second equation = {{{ 11y + 6z = 24 }}}
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{{{ y-4z=(-16) }}}
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{{{ 11y + 6z = 24 }}}
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We will now need to solve for a letter again( we will solve "y" since it is the easiest to solve)
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First equation {{{ y-4z=(-16) }}}
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We will move (-4z) to the right side
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{{{ y-4z=(-16) }}} = {{{ y-4z + 4z =(-16) + 4z }}}
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{{{ y-4z + 4z =(-16) + 4z }}} = {{{ y = (-16) + 4z }}}
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{{{ y = (-16) + 4z }}} = {{{ y = 4z - 16 }}} ( rearranging the numbers)
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Our first answer = {{{ 4z - 16 }}}
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Our second equation {{{ 11y + 6z = 24 }}}
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We will move "6z" to the right side
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{{{ 11y + 6z = 24 }}} = {{{ 11y + 6z - 6z = 24 - 6z }}}
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{{{ 11y + 6z - 6z = 24 - 6z }}} = {{{ 11y = 24 - 6z }}}
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Rearranging {{{ 11y = 24 - 6z }}} = {{{ 11y =  (-6z) + 24 }}}
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We will divide each side by "11"
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{{{ 11y =  (-6z) + 24 }}} = {{{ 11y/11 =  ((-6z) + 24)/11 }}}
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{{{ 11y/11 =  ((-6z) + 24)/11 }}} = {{{ cross (11)y/cross (11) =  ((-6z) + 24)/11 }}}
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{{{ cross (11)y/cross (11) =  ((-6z) + 24)/11 }}} = {{{ y =  ((-6z) + 24)/11 }}}
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Our second answer is {{{ ((-6z) + 24)/11 }}}
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We can now put both of our answers into an equation( since "y" equals both of our answers)
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Answer 1 = {{{ 4z - 16 }}}
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Answer 2 = {{{ ((-6z) + 24)/11 }}}
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We can make our equation, it equals {{{ 4z - 16 = ((-6z) + 24)/11 }}}
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{{{ 4z - 16 = ((-6z) + 24)/11 }}} = {{{ (4z - 16)/1 = ((-6z) + 24)/11 }}}
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We will cross multiply to get rid of the fractions
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{{{ highlight(4z - 16)/1 = ((-6z) + 24)/highlight(11) }}}
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It becomes, {{{ 44z - 176 = (-6z) + 24 }}}
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We will move (-6z) to the left side
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{{{ 44z - 176 = (-6z) + 24 }}} = {{{ 44z + 6z - 176 = (-6z) + 6z + 24 }}}
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{{{ 44z + 6z - 176 = (-6z) + 6z + 24 }}} = {{{ 50z - 176 = 24 }}}
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We will move (-176) to the right side
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{{{ 50z - 176 = 24 }}} = {{{ 50z - 176 + 176 = 24 + 176 }}}
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{{{ 50z - 176 + 176 = 24 + 176 }}} = {{{ 50z = 200 }}}
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We will divide each side by "50"
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{{{ 50z = 200 }}} = {{{ 50z/50 = 200/50 }}}
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{{{ 50z/50 = 200/50 }}} = {{{ cross(50)z/cross(50) = 4 }}}
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{{{ cross(50)z/cross(50) = 4 }}} = {{{ z = 4 }}}
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We found that "z" = 4, we can replace "z" with "4" in one of our 2 system/variable equations
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{{{ y-4z=(-16) }}}
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{{{ 11y + 6z = 24 }}}
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We will use the first equation
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{{{ y-4z=(-16) }}} = {{{ y-4(4)=(-16) }}}
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{{{ y-4(4)=(-16) }}} = {{{ y - 16 = (-16) }}}
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We will move (-16) to the right
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{{{ y - 16 = (-16) }}} = {{{ y - 16 + 16 = (-16) + 16 }}}
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{{{ y - 16 + 16 = (-16) + 16 }}} = {{{ y = 0 }}}
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We found that "y" = "0"
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We can now find "x", we need to replace "y" and "z" in one of the three original equations
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y = 0
z = 4
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x+3y+5z=20
y-4z=(-16)
3x-2y+9z=36
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We will use the first equation, since it will be the easiest( we can't use the second equation)
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{{{ x+3y+5z=20 }}} = {{{ x+3(0)+5(4)=20 }}}
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{{{ x+3(0)+5(4)=20 }}} = {{{ x+0+20=20 }}}
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{{{ x+0+20=20 }}} = {{{ x+20=20 }}}
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We will move the "20" over to the right side
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{{{ x+20=20 }}} = {{{ x+20-20=20 - 20 }}}
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{{{ x=0 }}}
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We found that "x" = "0", we now have all 3 variables, "x","y", and "z"
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x = 0
y = 0
z = 4
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We can check by replacing the letters with numbers in our third equation
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{{{ 3x-2y+9z=36 }}} = {{{ 3(0)- 2(0) + 9(4) = 36 }}} 
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{{{ 3(0)- 2(0) + 9(4) = 36 }}} = {{{ 0 - 0 + 36 = 36 }}} 
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{{{ 0 - 0 + 36 = 36 }}} = {{{ 36 = 36 }}} ( True)
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x = 0
y = 0
z = 4
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The solution set = (x,y,z), our solution set = (0,0,4)
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Hope I helped, Levi