SOLUTION: use cramer's rule to solve each system of equations x-y=4 3x-5y=8

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Question 742822: use cramer's rule to solve each system of equations
x-y=4
3x-5y=8
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%281%2Ax%2B-1%2Ay=4%2C3%2Ax%2B-5%2Ay=8%29



First let A=%28matrix%282%2C2%2C1%2C-1%2C3%2C-5%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 4 and 8 which are highlighted here:
system%281%2Ax%2B-1%2Ay=highlight%284%29%2C3%2Ax%2B-5%2Ay=highlight%288%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%28-5%29-%28-1%29%283%29=-2. 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%284%29%2C-1%2Chighlight%288%29%2C-5%29%29



Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%284%29%28-5%29-%28-1%29%288%29=-12. 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=%28-12%29%2F%28-2%29=6



So the first solution is x=6




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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%2C1%2C-1%2C3%2C-5%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%284%29%2C3%2Chighlight%288%29%29%29



Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%281%29%288%29-%284%29%283%29=-4.



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=%28-4%29%2F%28-2%29=2



So the second solution is y=2




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Final Answer:




So the solutions are x=6 and y=2 giving the ordered pair (6, 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.