SOLUTION: Solve the given system of equations by determinants. 2x-2y-5=0 4x+4=3y

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Question 1099888: Solve the given system of equations by determinants.
2x-2y-5=0
4x+4=3y
Found 2 solutions by richwmiller, ikleyn:
Answer by richwmiller(17219)   (Show Source): You can put this solution on YOUR website!
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 2 variables







First let . 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 and which are highlighted here:




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 . Remember that the determinant of the 2x2 matrix is . If you need help with calculating the determinant of any two by two matrices, then check out this solver.



Notation note: denotes the determinant of the matrix A.



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



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 (since we're replacing the 'x' column so to speak).






Now compute the determinant of to get . Once again, remember that the determinant of the 2x2 matrix is



To find the first solution, simply divide the determinant of by the determinant of to get:



So the first solution is




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


We'll follow the same basic idea to find the other solution. Let's reset by letting 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 (since we're replacing the 'y' column in a way).






Now compute the determinant of to get .



To find the second solution, divide the determinant of by the determinant of to get:



So the second solution is




====================================================================================

Final Answer:




So the solutions are and giving the ordered pair (-23/2, -14)




Once again, Cramer's Rule is dependent on determinants. Take a look at this 2x2 Determinant Solver if you need more practice with determinants.
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
On determinants and how to use them for solving systems of 2 equations in 2 unknowns see the lessons
    - What is a matrix?,
    - Determinant of a 2x2-matrix,
    - HOW TO solve system of linear equations in two unknowns using determinant (Cramer's rule),
    - Solving systems of linear equations in two unknowns using the Cramer's rule,
    - Solving word problems by the Cramer's rule after reducing to systems of linear equations in two unknowns,
in this site.

Also, you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic
     "2x2-Matrices, determinants, Cramer's rule for systems in two unknowns"


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.



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