+1833 \(Solve the system of equations \begin{align*} 5x+3z & = 1, \\ -x + y + z & = 0, \\ 3y + 2z &= 5. \end{align*} If there is a solution, write it as an ordered triple of integers or common fractions in simplest form. For example, if $x=1$, $y=\frac64$, and $z=-4$ were a solution, you would enter (1,3/2,-4). If there is no solution, enter the word "none" as your answer.\)
+128732 5x + 3z = 1 (1)
-x + y + z = 0 (2)
3y + 2z= 5 (3)
Uising (1) z = [ 1 - 5x] /3 (4)
Using (3) z = [5 - 3y] / 2 (5)
Equating (4) and (5) we have
[1 - 5x]/ 3 = [ 5-3y ] / 2 cross-multiply
2[ 1 - 5x] = 3 [ 5 - 3y] simplify
2 - 10x = 15 - 9y
9y = 13 + 10x
y = [13 + 10x]/ 9 (6)
Subbing (4) and (6) into (2), we have
-x + [13 + 10x]/9 + [1 -5x]/3 = 0
-9x + 13 + 10x + 3 - 15x = 0
-14x = -16
x = 8/7
And
y = [13 + 10(8/7)]/ 9 = 19/7
And
z = [ 1 - 5(8/7)] /3 = - 11/7
So {x, y, z} = { 8/7, 19/7, -11/7}
+23247 Solve:
5x + 3z = 1
-x + y + z = 0
3y + 2z = 5
This is a nasty problem -- if you have a calculator that handles matrices, it would be easy to solve.
If you have to do it by hand:
Combine the first and the second equations:
5x + 3z = 1 ---> 5x + 3z = 1
-x + y + z = 0 ---> multiply by 5 ---> -5x + 5y + 5z = 0
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Add down the columns: 5y + 8z = 1 ---> 5y + 8z = 1
Combine this answer with the third equation: 3y + 2z = 5 ---> multiply by -4 ---> -12y - 8z = -20
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Add down the columns: -7y = -19
Divide by -7: y = 19/7
Take this value for y and substitute it into the third equation:
3y + 2z = 5 ---> 3(19/7) + 2z = 5 ---> 57/7 + 2z = 5 ---> 2z = 5 - 57/7
---> 2z = -22/7
Divide by 2: z = -11/7
Take this value for z and substitute it into the first equation:
5x + 3z = 1 ---> 5x + 3(-11/7) = 1 ---> 5x -33/7 = 1 ---> 5x = 1 + 33/7
---> 5x = 40/7
Divide by 5: x = 8/7
+128732 5x + 3z = 1 (1)
-x + y + z = 0 (2)
3y + 2z= 5 (3)
Uising (1) z = [ 1 - 5x] /3 (4)
Using (3) z = [5 - 3y] / 2 (5)
Equating (4) and (5) we have
[1 - 5x]/ 3 = [ 5-3y ] / 2 cross-multiply
2[ 1 - 5x] = 3 [ 5 - 3y] simplify
2 - 10x = 15 - 9y
9y = 13 + 10x
y = [13 + 10x]/ 9 (6)
Subbing (4) and (6) into (2), we have
-x + [13 + 10x]/9 + [1 -5x]/3 = 0
-9x + 13 + 10x + 3 - 15x = 0
-14x = -16
x = 8/7
And
y = [13 + 10(8/7)]/ 9 = 19/7
And
z = [ 1 - 5(8/7)] /3 = - 11/7
So {x, y, z} = { 8/7, 19/7, -11/7}
