SOLUTION: Evaluate Dx -4x+7y=1 25=2x+5y

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Question 465149: Evaluate Dx
-4x+7y=1
25=2x+5y
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
Hi,
-4x+7y= 1
2x+5y=25


D%5Bx%5D = %28matrix%282%2C2%2C1%2C7%2C25%2C5%29%29 |Note: referred to as A%5Bx%5D below
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 2 variables



system%28-4%2Ax%2B7%2Ay=1%2C2%2Ax%2B5%2Ay=25%29



First let A=%28matrix%282%2C2%2C-4%2C7%2C2%2C5%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 25 which are highlighted here:
system%28-4%2Ax%2B7%2Ay=highlight%281%29%2C2%2Ax%2B5%2Ay=highlight%2825%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=%28-4%29%285%29-%287%29%282%29=-34. 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%2C7%2Chighlight%2825%29%2C5%29%29



Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%281%29%285%29-%287%29%2825%29=-170. 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=%28-170%29%2F%28-34%29=5



So the first solution is x=5




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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%2C-4%2C7%2C2%2C5%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%2C-4%2Chighlight%281%29%2C2%2Chighlight%2825%29%29%29



Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%28-4%29%2825%29-%281%29%282%29=-102.



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=%28-102%29%2F%28-34%29=3



So the second solution is y=3




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Final Answer:




So the solutions are x=5 and y=3 giving the ordered pair (5, 3)




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