SOLUTION: 5. Using Cramer's rule solve the following equations: (i) x+y-2 = 0 2x-y+2 = 3 4x+2y-22 = 2 (ii) x+2y = 9 2x-3y=4

Algebra ->  Matrices-and-determiminant -> SOLUTION: 5. Using Cramer's rule solve the following equations: (i) x+y-2 = 0 2x-y+2 = 3 4x+2y-22 = 2 (ii) x+2y = 9 2x-3y=4       Log On


   

Question 856461: 5. Using Cramer's rule solve the following equations:
(i) x+y-2 = 0
2x-y+2 = 3
4x+2y-22 = 2
(ii) x+2y = 9
2x-3y=4

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 3 variables



system%281%2Ax%2B1%2Ay%2B-2%2Az=0%2C2%2Ax%2B-1%2Ay%2B2%2Az=3%2C4%2Ax%2B2%2Ay%2B-22%2Az=2%29



First let A=%28matrix%283%2C3%2C1%2C1%2C-2%2C2%2C-1%2C2%2C4%2C2%2C-22%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 0, 3, and 2 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=54. 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=54. 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=%2854%29%2F%2854%29=1



So the first solution is x=1




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


We'll follow the same basic idea to find the other two solutions. Let's reset by letting A=%28matrix%283%2C3%2C1%2C1%2C-2%2C2%2C-1%2C2%2C4%2C2%2C-22%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=-54.



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=%28-54%29%2F%2854%29=-1



So the second solution is y=-1




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





Let's reset again by letting A=%28matrix%283%2C3%2C1%2C1%2C-2%2C2%2C-1%2C2%2C4%2C2%2C-22%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=0.



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=%280%29%2F%2854%29=0



So the third solution is z=0




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

Final Answer:




So the three solutions are x=1, y=-1, and z=0 giving the ordered triple (1, -1, 0)




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.



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



system%281%2Ax%2B2%2Ay=9%2C2%2Ax%2B-3%2Ay=4%29



First let A=%28matrix%282%2C2%2C1%2C2%2C2%2C-3%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 9 and 4 which are highlighted here:
system%281%2Ax%2B2%2Ay=highlight%289%29%2C2%2Ax%2B-3%2Ay=highlight%284%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=%281%29%28-3%29-%282%29%282%29=-7. 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%289%29%2C2%2Chighlight%284%29%2C-3%29%29



Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%289%29%28-3%29-%282%29%284%29=-35. 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=%28-35%29%2F%28-7%29=5



So the first solution is x=5




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


We'll follow the same basic idea to find the other solution. Let's reset by letting A=%28matrix%282%2C2%2C1%2C2%2C2%2C-3%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%2C1%2Chighlight%289%29%2C2%2Chighlight%284%29%29%29



Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%281%29%284%29-%289%29%282%29=-14.



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=%28-14%29%2F%28-7%29=2



So the second solution is y=2




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

Final Answer:




So the solutions are x=5 and y=2 giving the ordered pair (5, 2)




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