SOLUTION: What is the solution of the system? Solving using matrices. -7x+2y=-38 13x-3y=32 A. [-54] [-10] B. No Solution C. [10] [54] D. [-10] [-54]

Algebra ->  Matrices-and-determiminant -> SOLUTION: What is the solution of the system? Solving using matrices. -7x+2y=-38 13x-3y=32 A. [-54] [-10] B. No Solution C. [10] [54] D. [-10] [-54]      Log On


   

Question 871647: What is the solution of the system? Solving using matrices.
-7x+2y=-38
13x-3y=32
A.
[-54]
[-10]
B.
No Solution
C.
[10]
[54]
D.
[-10]
[-54]
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
(-10, -54) D.
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 2 variables



system%28-7%2Ax%2B2%2Ay=-38%2C13%2Ax%2B-3%2Ay=32%29



First let A=%28matrix%282%2C2%2C-7%2C2%2C13%2C-3%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 -38 and 32 which are highlighted here:
system%28-7%2Ax%2B2%2Ay=highlight%28-38%29%2C13%2Ax%2B-3%2Ay=highlight%2832%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=%28-7%29%28-3%29-%282%29%2813%29=-5. 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%28-38%29%2C2%2Chighlight%2832%29%2C-3%29%29



Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%28-38%29%28-3%29-%282%29%2832%29=50. 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=%2850%29%2F%28-5%29=-10



So the first solution is x=-10




---------------------------------------------------------


We'll follow the same basic idea to find the other solution. Let's reset by letting A=%28matrix%282%2C2%2C-7%2C2%2C13%2C-3%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-7%2Chighlight%28-38%29%2C13%2Chighlight%2832%29%29%29



Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%28-7%29%2832%29-%28-38%29%2813%29=270.



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=%28270%29%2F%28-5%29=-54



So the second solution is y=-54




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Final Answer:




So the solutions are x=-10 and y=-54 giving the ordered pair (-10, -54)




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