SOLUTION: [235] [432]

Algebra ->  Matrices-and-determiminant -> SOLUTION: [235] [432]      Log On


   

Question 867183: [235]
[432]
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
what is it You wanted to do?
IF...
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 2 variables



system%282%2Ax%2B3%2Ay=5%2C4%2Ax%2B3%2Ay=2%29



First let A=%28matrix%282%2C2%2C2%2C3%2C4%2C3%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 and 2 which are highlighted here:
system%282%2Ax%2B3%2Ay=highlight%285%29%2C4%2Ax%2B3%2Ay=highlight%282%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=%282%29%283%29-%283%29%284%29=-6. 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%285%29%2C3%2Chighlight%282%29%2C3%29%29



Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%285%29%283%29-%283%29%282%29=9. 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=%289%29%2F%28-6%29=-3%2F2



So the first solution is x=-3%2F2




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


We'll follow the same basic idea to find the other solution. Let's reset by letting A=%28matrix%282%2C2%2C2%2C3%2C4%2C3%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%2C2%2Chighlight%285%29%2C4%2Chighlight%282%29%29%29



Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%282%29%282%29-%285%29%284%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%28-6%29=8%2F3



So the second solution is y=8%2F3




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

Final Answer:




So the solutions are x=-3%2F2 and y=8%2F3 giving the ordered pair (-3/2, 8/3)




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