Use a matrix equation to solve the system of equations.
6x + 2y = 11
3x - 8y =1
My textbook gave this example, but explained it in a really weird way and I don't understand how to do it. Can someone help?
6x + 2y = 11
3x - 8y =1
The matrix equation is
AX=B
where A =
(6 2)
(3 -8)
X =
(x)
(y)
and B =
(11)
(1)
Since AX=B, if you multiply on the left by A^-1, you get
A^-1 * A X = A^-1 B
So, find A^-1, multiply by B and you end up with a 2x1 matrix which is the solution. See here:
www.wolframalpha.com/input/?i=%7B%7B6,2%7D,%7B3,-8%7D%7D*%7B%7Bx%7D,%7By%7D%7D+%3D+%7B%7B11%7D,%7B1%7D%7D
If they expect you to solve this using an inverse matrix, they must have shown
you how to find that matrix.
The last link I gave you from Wolfram showed what that matrix is.
There is a quick way to find the inverse of a 2 by 2
start with
a b
c d
find the determinant D, you said you knew how: D = (ad - bc)
inverse is
d/D -b/D
-c/D a/D
or
(1/D) times
d -b
-c a
in your case that becomes in reduced form
4/27 1/27
1/18 -1/9
What method was used?
Did they show the inverse matrix?
Did they use something called row-reduced echelon form?
(I would never think of using matrices for a system this simple)
btw, Wolfram also let's you find the inverse matrix
https://www.wolframalpha.com/input/?i=inverse+%7B%7B6,2%7D,%7B3,-8%7D%7D
I understand how to find the determinant, but my textbook also showed a part about how to get the solution for x and y, which I didn't understand. It said to multiply each side of the matrix equation by the inverse matrix, although I didn't understand what that meant and couldn't figure out how they did it. It says that the solution is (5/3, 1/2)
Sure, I can help you with that. To solve a system of equations using a matrix equation, we can represent the system of equations using matrices and then use matrix operations to solve for the variables.
Let's denote the variables as x and y, and rewrite the system of equations in matrix form. We'll represent the coefficients of the variables as a matrix A, the variables as a column vector X, and the constants as a column vector B.
The matrix equation for the given system of equations can be written as:
AX = B
where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
For the given system:
6x + 2y = 11
3x - 8y = 1
The coefficient matrix A is:
A = [[6, 2], [3, -8]]
The variable matrix X is:
X = [[x], [y]]
The constant matrix B is:
B = [[11], [1]]
Now, to solve for X, we can use matrix operations. The solution can be obtained by multiplying both sides of the equation by the inverse of A:
A^(-1)AX = A^(-1)B
Since the inverse of a matrix exists only if the matrix is non-singular (has a non-zero determinant), we need to check if matrix A is invertible.
To find the inverse of A, we calculate the determinant of A:
det(A) = (6 * -8) - (2 * 3) = (-48) - (6) = -54
Since the determinant is non-zero, matrix A is invertible, and we can proceed with finding the inverse.
Next, we find the inverse of A by using the inverse matrix formula:
A^(-1) = (1/det(A)) * adj(A)
Where adj(A) is the adjugate matrix of A.
The adjugate of A is found by swapping the diagonal elements and changing the sign of the off-diagonal elements:
adj(A) = [[-8, -2], [-3, 6]]
Now, we can find the inverse of A:
A^(-1) = (1/(-54)) * [[-8, -2], [-3, 6]] = [[4/27, 1/27], [1/18, -2/27]]
Finally, to find X, we multiply both sides of the equation by A^(-1):
X = A^(-1)B = [[4/27, 1/27], [1/18, -2/27]] * [[11], [1]]
Multiplying the matrices, we get:
X = [[(4/27 * 11 + 1/27 * 1)], [(1/18 * 11 - 2/27 * 1)]]
Simplifying further:
X = [[47/27], [11/18]]
Therefore, the solution to the system of equations is x = 47/27 and y = 11/18.