SOLUTION: Hello, my name is steve and i would like you to help me solve the Equation 7x+3y=21 by cramers rule if you would please thank you. 5x-4y=9

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Question 610284: Hello, my name is steve and i would like you to help me solve the Equation
7x+3y=21 by cramers rule if you would please thank you.
5x-4y=9
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
For more help with solving systems of linear equations using cramers rule, check out this solver.


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



system%287%2Ax%2B3%2Ay=21%2C5%2Ax%2B-4%2Ay=9%29



First let A=%28matrix%282%2C2%2C7%2C3%2C5%2C-4%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 21 and 9 which are highlighted here:
system%287%2Ax%2B3%2Ay=highlight%2821%29%2C5%2Ax%2B-4%2Ay=highlight%289%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=%287%29%28-4%29-%283%29%285%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%2821%29%2C3%2Chighlight%289%29%2C-4%29%29



Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%2821%29%28-4%29-%283%29%289%29=-111. 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=%28-111%29%2F%28-43%29=111%2F43



So the first solution is x=111%2F43




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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%2C7%2C3%2C5%2C-4%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%2C7%2Chighlight%2821%29%2C5%2Chighlight%289%29%29%29



Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%287%29%289%29-%2821%29%285%29=-42.



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=%28-42%29%2F%28-43%29=42%2F43



So the second solution is y=42%2F43




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Final Answer:




So the solutions are x=111%2F43 and y=42%2F43 giving the ordered pair (111/43, 42/43)




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







For more help with solving systems of linear equations using cramers rule, check out this solver.