SOLUTION: Solve each system of equations: x+2y=12 3y-4z=25 x+6y+z=20

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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
.
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,
.
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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, 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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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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