SOLUTION: Using cramers rule: 6x+2y=-1 and -x+10y=5 What are the values of D,Dx,Dy

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Question 610807: Using cramers rule: 6x+2y=-1 and -x+10y=5
What are the values of D,Dx,Dy
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%286%2Ax%2B2%2Ay=-1%2C-1%2Ax%2B10%2Ay=5%29



First let A=%28matrix%282%2C2%2C6%2C2%2C-1%2C10%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%2B2%2Ay=highlight%28-1%29%2C-1%2Ax%2B10%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%2810%29-%282%29%28-1%29=62. 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%28-1%29%2C2%2Chighlight%285%29%2C10%29%29



Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%28-1%29%2810%29-%282%29%285%29=-20. 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=%28-20%29%2F%2862%29=-10%2F31



So the first solution is x=-10%2F31




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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%2C2%2C-1%2C10%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%28-1%29%2C-1%2Chighlight%285%29%29%29



Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%286%29%285%29-%28-1%29%28-1%29=29.



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=%2829%29%2F%2862%29=29%2F62



So the second solution is y=29%2F62




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Final Answer:




So the solutions are x=-10%2F31 and y=29%2F62 giving the ordered pair (-10/31, 29/62)




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