SOLUTION: solve the following system of equations X – 3z = -5 2x- y + 2z= 16 7x – 3y -5z =19 can you walk me thru the steps please

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Question 634643: solve the following system of equations
X – 3z = -5
2x- y + 2z= 16
7x – 3y -5z =19
can you walk me thru the steps please
Found 3 solutions by ankor@dixie-net.com, radh, ewatrrr:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
solve the following system of equations
X – 3z = -5
2x- y + 2z= 16
7x – 3y -5z = 19
There is more than one way to solve this, but this is a pretty simple one.
A combination of substitution and elimination.
:
Rearrange the 1st equation to use for substitution:
x = (3z-5)
:
Multiply the 2nd equation by 3, subtract from the 3rd equation
7x – 3y - 5z = 19
6x - 3y + 6z = 48
-------------------
x + 0 - 11z = -29
Replace x with (3z-5) from the 1st equation
(3z-5) - 11z =-29
3z - 11z = -29 + 5
-8z = - 24
z = %28-24%29%2F%28-8%29
z = +3
:
Back to the 1st equation, replace z with 3
x = 3(3) - 5
x = 9 - 5
x = 4
:
Use the original 2nd equation to find y, Replace x and z
2x - y + 2z = 16
2(4) - y + 2(3) = 16
8 - y + 6 = 16
-y = 16 - 8 - 6
-y = 2
Y has to be positive, multiply both sides by -1
y = -2
:
Our Solution: x=4; y=-2; z=3
:
:
Check these in the 3rd original equation
7x – 3y -5z = 19
7(4) - 3(-2) - 5(3) =
28 + 6 - 15 = 19; confirms our solutions

Answer by radh(108) About Me  (Show Source):
You can put this solution on YOUR website!
This should help:

Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 3 variables







First let A=%28matrix%283%2C3%2C1%2C0%2C-3%2C2%2C-1%2C2%2C7%2C-3%2C-5%29%29. This is the matrix formed by the coefficients of the given system of equations.


Take note that the right hand values of the system are -5, 16, and 19 and they are highlighted here:




These values are important as they will be used to replace the columns of the matrix A.




Now let's calculate the the determinant of the matrix A to get abs%28A%29=8. To save space, I'm not showing the calculations for the determinant. However, if you need help with calculating the determinant of the matrix A, check out this solver.



Notation note: abs%28A%29 denotes the determinant of the matrix A.



---------------------------------------------------------



Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5Bx%5D (since we're replacing the 'x' column so to speak).






Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=32. Again, as a space saver, I didn't include the calculations of the determinant. Check out this solver to see how to find this determinant.



To find the first solution, simply divide the determinant of A%5Bx%5D by the determinant of A to get: x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%2832%29%2F%288%29=4



So the first solution is x=4




---------------------------------------------------------


We'll follow the same basic idea to find the other two solutions. Let's reset by letting A=%28matrix%283%2C3%2C1%2C0%2C-3%2C2%2C-1%2C2%2C7%2C-3%2C-5%29%29 again (this is the coefficient matrix).




Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5By%5D (since we're replacing the 'y' column in a way).






Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=-16.



To find the second solution, divide the determinant of A%5By%5D by the determinant of A to get: y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%28-16%29%2F%288%29=-2



So the second solution is y=-2




---------------------------------------------------------





Let's reset again by letting A=%28matrix%283%2C3%2C1%2C0%2C-3%2C2%2C-1%2C2%2C7%2C-3%2C-5%29%29 which is the coefficient matrix.



Replace the third column of A (that corresponds to the variable 'z') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5Bz%5D






Now compute the determinant of A%5Bz%5D to get abs%28A%5Bz%5D%29=24.



To find the third solution, divide the determinant of A%5Bz%5D by the determinant of A to get: z=%28abs%28A%5Bz%5D%29%29%2F%28abs%28A%29%29=%2824%29%2F%288%29=3



So the third solution is z=3




====================================================================================

Final Answer:




So the three solutions are x=4, y=-2, and z=3 giving the ordered triple (4, -2, 3)




Note: there is a lot of work that is hidden in finding the determinants. Take a look at this 3x3 Determinant Solver to see how to get each determinant.



:)


Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
 

Hi,

 x- 3z = -5    | x = 3z- 5

Have a relationship between x and z, so...eliminating y from the other two EQs

2x- y + 2z= 16      |mulitplying 1st by 3 and subtracting it form the 2nd EQ

7x – 3y -5z =19

   x  - 11z =  -29 

 (3z- 5 - 11z = -29

         -8z = -24

           z = 3  and x = 4  and y = +2%2A4+%2B+2%2A3+-+16+=+-2