Question 167587This question is from textbook Algebra 2
: Solve each system of equations:
x+2y=12
3y-4z=25
x+6y+z=20
This question is from textbook Algebra 2
Answer by Electrified_Levi(103)   (Show Source):
You can put this solution on YOUR website! Hi, Hope I can help
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Solve each system of equations:
x+2y=12
3y-4z=25
x+6y+z=20
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There is no fast way to solving these problems, but I will show you, what I believe is the easiest way, how to solve these problems
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First we have to solve for a variable, we will need to solve for "y" in each equation, since they all have a variable "y" ( It usually doesn't matter, but in this case it does.
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First equation, x+2y=12. (solve for "y")
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 , We can solve for "y", first we will move the "x" to the right side
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 =  =  , we can rearrange the right side
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=  , to solve for "y" we will divide each side by "2"
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=  = 
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 is our first answer
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Second equation solve for "y", 3y-4z=25
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 , we need to move (-4z) to the right side
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=  =  , we can rearrange the right side
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=  , to solve for "y" we will need to divide each side by "3"
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=  = 
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 is your second answer
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Now let us solve the last equation for "y", x+6y+z=20
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 , first we will move the "z" to the right side
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=  =  , now let's move the "x" to the right side
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=  ,  , let us rearrange the right side
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=  , to solve for "y", we will divide both sides by "6"
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=  = 
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 is our third answer
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Now let us put all of our answers side by side
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First,
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Second, 
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Third,
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Since all three answers are equal to "y" all of them are equal to each other, first we have to have both "x" and "z" in all equations
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First, , this has no "z's" so this equation would be 
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Second, , this has no "x's" so this equation would be 
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Third, , this already has both "x" and "z"
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Here are the new equations
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First,
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Second, 
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Third,
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All of them equal each other, first we will let the first two equations equal each other
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 , we will use cross multiplication
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=  = 
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 , we will rearrange the left side
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=  , now we can use the distribution method to solve even more,
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=  =  = 
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Don't forget to use the positive, and negative signs
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 , we will move the (-3x) and "0z" to the right side
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=  =  , now we will move the "50" to the left side
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=  =  , rearranging we get
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= 
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is our first answer
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Now we can use the last two equations ( we can use the middle equation twice
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Second,
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Third,
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 , we can use cross multiplication
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=  = 
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 , rearranging the left side
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=  , we will use the distrubution method,
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 =  = 
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Remember the signs,  , we will move "-3x" to the left side
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=  =  , we will now move "-3z to the left side
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=  =  , we will lastly move "150" to the right side
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=  = 
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is the second answer, once again let us put our two answers side by side
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First,
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Second, , we can reduce this equation by dividing each side by "3", =  =  = 
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Our new equations are
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First,
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Second,
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Now we need to solve for a variable again, doesn't matter which one, we will solve for "x"
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First, , we need to move "8z" to the right side,
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=  = 
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Rearranging the right side, =  , we can solve for "x" by dividing each side by "3"
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=  = 
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 is our first answer
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We can now solve for "x" in the second equation,
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, we will move "9z" to the right side
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=  =  , rearranging the right side, = 
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 is our second answer, let us put our two answers together
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First,
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Second,
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Since our two answers equal "x" they will equal each other
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 =  , using cross-multiplication
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 = 
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 , rearranging the left side
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=  , using distribution
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 = 
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Remember the signs,  , we will now solve for "z", we will move (-27z) to the left side
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=  =  , now we will move (-14) to the right side
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=  =  , to solve "z" we will divide each side by "19"
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=  =  , to check our answer we will replace "z" with (-4) in our equation
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=  =  =  =  (True)
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Now that we know that  , we can replace it in one of the equations with two variables
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First,
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Second,
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We will use the second equation, replace "z" with (-4)
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=  =  =  , lets move (-36) to the right side
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=  = 
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Let's check our answer by replacing "x" with "6", and "z" with (-4) in the equation,
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=  =  =  =  (True)
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"x" = "6"
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"z" = (-4),
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Now we can find "y" by replacing "x" and "y" in one of the three original equations
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x+2y=12
3y-4z=25
x+6y+z=20
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We will need to use the third equation, replace "x" with "6", "z" with (-4)
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=  =  =  =  , we will move "2" to the right side
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=  =  , to find "y", divide each side by "6"
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=  =  , we can check by replacing "x" with "6", "y" with "3", "z" with (-4) in the equation
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=  =  =  =  ( True )
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x = 6
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y = 3
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z = (-4)
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We can check by replacing the other two original equations
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=  =  =  (True)
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=  =  =  (True)
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x = 6
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y = 3
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z = (-4)
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ordered pairs are given as (x,y,z), our ordered pair = ( 6,3,-4)
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Hope I helped, Levi
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