Solve the system of equations using matrices.
2x + 5y = 19
3x + 4y = 11
and also:
is the system consistent, inconsistent, or neither?
is the system dependent, independent, or neither?
is the solution a line, a point, a plane, or is there no solution?
Hard to do matrices here, but if we enclose each row in braces, we have
{{2,5},{3,4}}*{{x},{y}} = {{19},{11}}
so x = -3, y = 5
To solve the system of equations using matrices, we can represent the system in matrix form. Let's call the coefficient matrix A, the variable matrix X, and the constant matrix B. The equation would be AX = B.
First, let's represent the given system of equations in matrix form:
2x + 5y = 19 (equation 1)
3x + 4y = 11 (equation 2)
The coefficient matrix A is:
A = [2 5]
[3 4]
The variable matrix X is:
X = [x]
[y]
And the constant matrix B is:
B = [19]
[11]
Now, we have the equation AX = B. To solve for X, we can find the inverse of matrix A and multiply it by B:
X = A^(-1) * B
To find the inverse of matrix A, we can use the formula:
A^(-1) = 1/det(A) * adj(A)
where det(A) is the determinant of matrix A and adj(A) is the adjugate of matrix A.
The determinant of matrix A is calculated as:
det(A) = (2 * 4) - (3 * 5) = -7
So, the inverse of matrix A is:
A^(-1) = (-1/7) * [4 -5]
[-3 2]
Now, we can multiply A^(-1) by B to solve for X:
X = A^(-1) * B
= (-1/7) * [4 -5] * [19]
[11]
Multiplying the matrices, we get:
X = [-1/7 * (4 * 19 + (-5) * 11)]
[-1/7 * (-3 * 19 + 2 * 11)]
Simplifying further:
X = [-1/7 * (76 - 55)]
[-1/7 * (-57 + 22)]
X = [-1/7 * 21]
[-1/7 * (-35)]
X = [-3]
[5]
Therefore, the solution to the system of equations is x = -3 and y = 5.