SOLUTION: Solve by using matrx 2x-y=-2 3x+4y=3

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Question 821785: Solve by using matrx
2x-y=-2
3x+4y=3
Found 2 solutions by jsmallt9, ewatrrr:
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
There are several matrix-based methods for solving such a system. Please re-post your problem and specify which method should be used. (If "any" method may be used, then say so in your re-post and list the methods you have learned so the tutor can use a method you would have a chance of understanding.)

Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
 

Hi,

Solve by using matrx

2x-y=-2

3x+4y=3

x= -5⁄11 and y= 12⁄11   giving the ordered pair (-5/11, 12/11)




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




  

  

  system%282%2Ax%2B-1%2Ay=-2%2C3%2Ax%2B4%2Ay=3%29

  

  

  

  First let A=%28matrix%282%2C2%2C2%2C-1%2C3%2C4%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 -2 and 3 which are highlighted here: 

  system%282%2Ax%2B-1%2Ay=highlight%28-2%29%2C3%2Ax%2B4%2Ay=highlight%283%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=%282%29%284%29-%28-1%29%283%29=11. 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.

  

  

  

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

  

  

  

  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%28-2%29%2C-1%2Chighlight%283%29%2C4%29%29

  

  

  

  Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%28-2%29%284%29-%28-1%29%283%29=-5. 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=%28-5%29%2F%2811%29=-5%2F11

  

  

  

  So the first solution is x=-5%2F11

  

  

  

  

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

  

  

  We'll follow the same basic idea to find the other solution. Let's reset by letting A=%28matrix%282%2C2%2C2%2C-1%2C3%2C4%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%2C2%2Chighlight%28-2%29%2C3%2Chighlight%283%29%29%29

  

  

  

  Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%282%29%283%29-%28-2%29%283%29=12.

  

  

  

  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=%2812%29%2F%2811%29=12%2F11

  

  

  

  So the second solution is y=12%2F11

  

  

  

  

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

  

  Final Answer:

  

  

  

  

  So the solutions are x=-5%2F11 and y=12%2F11 giving the ordered pair (-5/11, 12/11)

  

  

  

  

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