SOLUTION: solve 2x+y=3 and 3x-2y=8 using matrices

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Question 886263: solve 2x+y=3 and 3x-2y=8 using matrices

Found 2 solutions by josgarithmetic, richwmiller:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
%28matrix%282%2C3%2C2%2C1%2C3%2C3%2C-2%2C8%29%29

%28matrix%282%2C3%2C2%2C1%2C3%2C0%2C-5%2C-1%29%29, R2-3*R1

%28matrix%282%2C3%2C2%2C1%2C3%2C0%2C-1%2C-1%2F5%29%29, (1/5)*R2

%28matrix%282%2C3%2C2%2C0%2C4%2F5%2C0%2C-1%2C-1%2F5%29%29, R1=R1+R2

%28matrix%282%2C3%2C1%2C0%2C2%2F5%2C0%2C1%2C1%2F5%29%29
The solution matrix

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
The other tutor is wrong.
using Gauss Jordan
divide row 1 by 2/1
1,1/2,3/2
3,-2,8


add down (-3/1) *row 1 to row 2
1,1/2,3/2
0,-7/2,7/2


divide row 2 by -7/2
1,1/2,3/2
0,1,-1


add up (-1/2) *row 2 to row 1
1,0,2
0,1,-1

final
1,0,2
0,1,-1
x=2
y=-1

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



system%282%2Ax%2B1%2Ay=3%2C3%2Ax%2B-2%2Ay=8%29



First let A=%28matrix%282%2C2%2C2%2C1%2C3%2C-2%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 3 and 8 which are highlighted here:
system%282%2Ax%2B1%2Ay=highlight%283%29%2C3%2Ax%2B-2%2Ay=highlight%288%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%28-2%29-%281%29%283%29=-7. 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%283%29%2C1%2Chighlight%288%29%2C-2%29%29



Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%283%29%28-2%29-%281%29%288%29=-14. 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=%28-14%29%2F%28-7%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%2C2%2C1%2C3%2C-2%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%283%29%2C3%2Chighlight%288%29%29%29



Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%282%29%288%29-%283%29%283%29=7.



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=%287%29%2F%28-7%29=-1



So the second solution is y=-1




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Final Answer:




So the solutions are x=2 and y=-1 giving the ordered pair (2, -1)




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