SOLUTION: solve x&y by using determinant method 4x - 8y =4 6x + 20y =- 2 4,-8,4 6,20,-2

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Question 889290: solve x&y by using determinant method
4x - 8y =4
6x + 20y =- 2
4,-8,4
6,20,-2
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
determinant = 128
x = 64/128 = 1/2 = 0.5
y = -32/128 = -1/4 = -0.25
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 2 variables



system%284%2Ax%2B-8%2Ay=4%2C6%2Ax%2B20%2Ay=-2%29



First let A=%28matrix%282%2C2%2C4%2C-8%2C6%2C20%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 -2 which are highlighted here:
system%284%2Ax%2B-8%2Ay=highlight%284%29%2C6%2Ax%2B20%2Ay=highlight%28-2%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=%284%29%2820%29-%28-8%29%286%29=128. 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-8%2Chighlight%28-2%29%2C20%29%29



Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%284%29%2820%29-%28-8%29%28-2%29=64. 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=%2864%29%2F%28128%29=1%2F2



So the first solution is x=1%2F2




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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%2C4%2C-8%2C6%2C20%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%2C4%2Chighlight%284%29%2C6%2Chighlight%28-2%29%29%29



Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%284%29%28-2%29-%284%29%286%29=-32.



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=%28-32%29%2F%28128%29=-1%2F4



So the second solution is y=-1%2F4




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Final Answer:




So the solutions are x=1%2F2 and y=-1%2F4 giving the ordered pair (1/2, -1/4)




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