SOLUTION: Use Cramer's rule to solve the systems: 1. 2x-9y-z=-72 x+3y+4z=51 -6x+y+z=-4 2. 4x+6y=14 2x+y=-3

Algebra ->  Matrices-and-determiminant -> SOLUTION: Use Cramer's rule to solve the systems: 1. 2x-9y-z=-72 x+3y+4z=51 -6x+y+z=-4 2. 4x+6y=14 2x+y=-3      Log On


   

Question 274496: Use Cramer's rule to solve the systems:
1. 2x-9y-z=-72
x+3y+4z=51
-6x+y+z=-4

2. 4x+6y=14
2x+y=-3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1

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







First let A=%28matrix%283%2C3%2C2%2C-9%2C-1%2C1%2C3%2C4%2C-6%2C1%2C1%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 -72, 51, and -4 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=204. 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=612. 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=%28612%29%2F%28204%29=3



So the first solution is x=3




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We'll follow the same basic idea to find the other two solutions. Let's reset by letting A=%28matrix%283%2C3%2C2%2C-9%2C-1%2C1%2C3%2C4%2C-6%2C1%2C1%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=1632.



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=%281632%29%2F%28204%29=8



So the second solution is y=8




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





Let's reset again by letting A=%28matrix%283%2C3%2C2%2C-9%2C-1%2C1%2C3%2C4%2C-6%2C1%2C1%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=1224.



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=%281224%29%2F%28204%29=6



So the third solution is z=6




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Final Answer:




So the three solutions are x=3, y=8, and z=6 giving the ordered triple (3, 8, 6)




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.





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# 2

Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 2 variables



system%284%2Ax%2B6%2Ay=14%2C2%2Ax%2B1%2Ay=-3%29



First let A=%28matrix%282%2C2%2C4%2C6%2C2%2C1%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 14 and -3 which are highlighted here:
system%284%2Ax%2B6%2Ay=highlight%2814%29%2C2%2Ax%2B1%2Ay=highlight%28-3%29%29



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=%284%29%281%29-%286%29%282%29=-8. Remember that the determinant of the 2x2 matrix A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 is abs%28A%29=ad-bc. If you need help with calculating the determinant of any two by two matrices, then 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).


A%5Bx%5D=%28matrix%282%2C2%2Chighlight%2814%29%2C6%2Chighlight%28-3%29%2C1%29%29



Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%2814%29%281%29-%286%29%28-3%29=32. Once again, remember that the determinant of the 2x2 matrix A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 is abs%28A%29=ad-bc



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%28-8%29=-4



So the first solution is x=-4




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


We'll follow the same basic idea to find the other solution. Let's reset by letting A=%28matrix%282%2C2%2C4%2C6%2C2%2C1%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).


A%5Bx%5D=%28matrix%282%2C2%2C4%2Chighlight%2814%29%2C2%2Chighlight%28-3%29%29%29



Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%284%29%28-3%29-%2814%29%282%29=-40.



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=%28-40%29%2F%28-8%29=5



So the second solution is y=5




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

Final Answer:




So the solutions are x=-4 and y=5 giving the ordered pair (-4, 5)




Once again, Cramer's Rule is dependent on determinants. Take a look at this 2x2 Determinant Solver if you need more practice with determinants.