SOLUTION: use cramer's rule to solve each system of equations 6x-5y=1 3x+2y=5

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Question 742825: use cramer's rule to solve each system of equations
6x-5y=1
3x+2y=5
Answer by MathLover1(20850) 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%286%2Ax%2B-5%2Ay=1%2C3%2Ax%2B2%2Ay=5%29



First let A=%28matrix%282%2C2%2C6%2C-5%2C3%2C2%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 1 and 5 which are highlighted here:
system%286%2Ax%2B-5%2Ay=highlight%281%29%2C3%2Ax%2B2%2Ay=highlight%285%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=%286%29%282%29-%28-5%29%283%29=27. 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%281%29%2C-5%2Chighlight%285%29%2C2%29%29



Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%281%29%282%29-%28-5%29%285%29=27. 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=%2827%29%2F%2827%29=1



So the first solution is x=1




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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%2C6%2C-5%2C3%2C2%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%2C6%2Chighlight%281%29%2C3%2Chighlight%285%29%29%29



Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%286%29%285%29-%281%29%283%29=27.



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=%2827%29%2F%2827%29=1



So the second solution is y=1




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Final Answer:




So the solutions are x=1 and y=1 giving the ordered pair (1, 1)




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