PLEASE help.
Solve the system of equations by first expressing it in matrix form as and then evaluating.
a). 3x-2y=5
4x-y =-10
b). 3x -2y =-2
4x -y = 3
To solve a system of equations, we can express it in matrix form and then evaluate it.
a) Let's solve the system of equations:
1. Write the system of equations in matrix form:
[3 -2] [x] [ 5]
[4 -1] [y] = [-10]
2. To solve this system, we can use matrix algebra. Let's denote the coefficient matrix as A, the variable matrix as X, and the constant matrix as B.
A = [3 -2]
[4 -1]
X = [x]
[y]
B = [ 5]
[-10]
3. Now we can write the equation AX = B. To find X, we can multiply both sides of the equation by the inverse of A, denoted as A^(-1):
A^(-1) * (AX) = A^(-1) * B
Since A^(-1) * A gives the identity matrix I, we have:
X = A^(-1) * B
4. Find the inverse of matrix A, denoted as A^(-1). Use the formula for finding the inverse of a 2x2 matrix:
A^(-1) = (1/(ad-bc)) * [d -b]
[-c a]
In our case:
a = 3, b = -2, c = 4, d = -1
A^(-1) = (1/((-1)(3) - (-2)(4))) * [ -1 2]
[ 4 -3]
Simplifying gives:
A^(-1) = (1/(-3 + 8)) * [ -1 2]
[ 4 -3]
A^(-1) = (1/5) * [ -1 2]
[ 4 -3]
5. Multiply A^(-1) by B to obtain the solution X:
X = A^(-1) * B
X = (1/5) * [ -1 2] * [ 5]
[-10]
Simplifying gives:
X = (1/5) * [ -5 + 20]
[ 20 + 30]
X = (1/5) * [ 15]
[ 50]
X = [ 3]
[10]
Therefore, the solution to the system of equations is x = 3 and y = 10.
b) Follow the same steps as in part a) to solve this system of equations. The matrix form of the system is:
[3 -2] [x] [-2]
[4 -1] [y] = [ 3]
Find A^(-1) and then multiply it by B to find the solution X.