SOLUTION: use crammer's rule to solve the system of equation: 2x+y=-2 and x-2y=19

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Question 751413: use crammer's rule to solve the system of equation: 2x+y=-2 and x-2y=19

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 2 variables



system%282%2Ax%2B1%2Ay=-2%2C1%2Ax%2B-2%2Ay=19%29



First let A=%28matrix%282%2C2%2C2%2C1%2C1%2C-2%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 -2 and 19 which are highlighted here:
system%282%2Ax%2B1%2Ay=highlight%28-2%29%2C1%2Ax%2B-2%2Ay=highlight%2819%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%28-2%29-%281%29%281%29=-5. 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.



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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%28-2%29%2C1%2Chighlight%2819%29%2C-2%29%29



Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%28-2%29%28-2%29-%281%29%2819%29=-15. 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=%28-15%29%2F%28-5%29=3



So the first solution is x=3




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We'll follow the same basic idea to find the other solution. Let's reset by letting A=%28matrix%282%2C2%2C2%2C1%2C1%2C-2%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%28-2%29%2C1%2Chighlight%2819%29%29%29



Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%282%29%2819%29-%28-2%29%281%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=%2840%29%2F%28-5%29=-8



So the second solution is y=-8




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Final Answer:




So the solutions are x=3 and y=-8 giving the ordered pair (3, -8)




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