SOLUTION: I need help solving the system 5X - 2Y = 14 4X + 7Y = -6

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Question 293092: I need help solving the system 5X - 2Y = 14
4X + 7Y = -6
Answer by Deina(147) About Me  (Show Source):
You can put this solution on YOUR website!
Thanks to Jim Thompson for writing this wonderful solver for this situation!

I will admit that, at first glance it looks a bit scary, but if you go point by point through it, making sure that you understand what he's doing, it's easy like cake!

Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 2 variables



system%285%2Ax%2B-2%2Ay=14%2C4%2Ax%2B7%2Ay=-6%29



First let A=%28matrix%282%2C2%2C5%2C-2%2C4%2C7%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 14 and -6 which are highlighted here:
system%285%2Ax%2B-2%2Ay=highlight%2814%29%2C4%2Ax%2B7%2Ay=highlight%28-6%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=%285%29%287%29-%28-2%29%284%29=43. 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%2814%29%2C-2%2Chighlight%28-6%29%2C7%29%29



Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%2814%29%287%29-%28-2%29%28-6%29=86. 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=%2886%29%2F%2843%29=2



So the first solution is x=2




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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%2C5%2C-2%2C4%2C7%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%2C5%2Chighlight%2814%29%2C4%2Chighlight%28-6%29%29%29



Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%285%29%28-6%29-%2814%29%284%29=-86.



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=%28-86%29%2F%2843%29=-2



So the second solution is y=-2




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Final Answer:




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



and bob's your uncle!