which product represents the solution to the system 3x+4y=-2 and 2x+5y=6
[3 2]*[-2
[4 5] [6]
1/7[3 2]*[-2]
[4 5] [6]
[5 -2]*[-2]
[-4 3] [6]
1/7[5 -4]*[-2]
[-2 3] [6]
could someone explain
your given system can be written in the following matrix form
[ 5-4]*[x]=[-2]
[-2 3] [y] [6]
the inverse of the first matrix is
1/17 of
[ 5-4]
[-2 3]
if you multiply the left side by that inverse you get the identity matrix
[1 0]
[0 1]
of course you would then also multiply the right side by that , so it looks like the last answer is the correct one
oops!!
that first matrix of course should have been
[ 3 4]*[x]=[-2]
[ 2 5] [y] [6]
I copied incorrectly from my paper
To find the solution to a system of equations, you can use the matrix equation AX = B, where A is the coefficient matrix of the variables, X is the column matrix of the variables, and B is the column matrix of the constants.
In the given system:
3x + 4y = -2 (equation 1)
2x + 5y = 6 (equation 2)
We can rewrite the system in matrix form as:
[3 4] [x] [-2]
[2 5] * [y] = [6]
To find the solution, we can solve for X by multiplying both sides of the equation by the inverse of the coefficient matrix A:
X = A^(-1) * B
To calculate the inverse of a 2x2 matrix, A^(-1), we use the equation:
A^(-1) = 1/(ad - bc) * [d -b]
[-c a]
For the given system:
A = [3 4]
[2 5]
Calculating the determinant (ad - bc), we get:
(3*5) - (4*2) = 15 - 8 = 7
The inverse of matrix A, A^(-1), is then:
A^(-1) = 1/7 * [5 -4]
[-2 3]
Finally, we can find X by multiplying the inverse of matrix A with matrix B:
X = A^(-1) * B
Plugging in the values, we get:
X = 1/7 * [5 -4] * [-2]
[-2 3] [6]
Calculating the matrix multiplication, we get:
X = 1/7 * [5*(-2) + (-4)*6]
[-2*(-2) + 3*6]
Simplifying, we find:
X = 1/7 * [-10 - 24]
[4 + 18]
X = [-34/7]
[22/7]
Therefore, the solution to the system is x = -34/7 and y = 22/7.