Use the matrix tool to solve the system of equations. Choose the correct ordered pair. -3x + 5y = 19
5x - 5y = -5
-3x + 5y = 19
5x - 5y = -5
-3 5 19
5 -5 -5
(7,8)
4x-5y=13
3x-2y=14
To solve the system of equations using the matrix tool, we can write the system of equations as a matrix equation.
First, let's rewrite the system of equations in the form Ax = b, where A is the coefficient matrix, x is the column matrix of variables, and b is the column matrix of constants.
The coefficient matrix A:
A = [[-3, 5], [5, -5]]
The variable matrix x:
x = [[x], [y]]
The constant matrix b:
b = [[19], [-5]]
Now, we can solve for x by multiplying both sides of the equation by the inverse of matrix A:
A^(-1) * A * x = A^(-1) * b
Since A^(-1) * A is the identity matrix, we have:
x = A^(-1) * b
To find the inverse of matrix A, we can use the formula for a 2x2 matrix:
A^(-1) = 1 / (ad - bc) * [[d, -b], [-c, a]],
where a = -3, b = 5, c = 5, and d = -5.
Now, let's calculate the inverse of matrix A:
A^(-1) = 1 / (ad - bc) * [[-5, -5], [-5, -3]] = [[-1/10, -1/10], [-1/10, -3/10]]
Next, let's substitute the values of A^(-1) and b into the equation x = A^(-1) * b and calculate the solution:
x = [[-1/10, -1/10], [-1/10, -3/10]] * [[19], [-5]]
= [[-1/10*19 + -1/10*(-5)], [-1/10*19 + -3/10*(-5)]]
= [[-2/10], [-6/10]]
= [[-1/5], [-3/5]]
Therefore, the correct ordered pair (x, y) is (-1/5, -3/5).
To solve the system of equations using the matrix tool, we need to represent the coefficients of the variables and the constants as a matrix. Let's call the matrix A and the matrix B:
Matrix A:
[ -3 5 ]
[ 5 -5 ]
Matrix B:
[ 19 ]
[ -5 ]
Next, we need to find the inverse of matrix A and multiply it by matrix B. This can be done as follows:
Inverse of A * B = A^(-1) * B
If the inverse of A exists, we can find it by using the formula:
A^(-1) = (1 / (ad - bc)) * Adj(A)
where Adj(A) denotes the adjugate of matrix A.
Calculating the adjugate of A:
Adj(A) = [ -5 -5 ]
[ -5 -3 ]
Calculating the determinate of A:
ad - bc = (-3 * -5) - (5 * 5) = 15 - 25 = -10
Since the determinant is not zero (-10 ≠ 0), the inverse of A exists.
Using the inverse of A, we can calculate the solution matrix X as follows:
X = A^(-1) * B
Multiplying A^(-1) and B:
X = (1 / (ad - bc)) * Adj(A) * B
Calculating (1 / (ad - bc)):
1 / (ad - bc) = 1 / -10 = -1/10
Now multiplying the matrices:
X = (-1/10) * [ -5 -5 ] * [ 19 ] = [ (-1/10) * ( -5 * 19 - -5 * -5 ) ]
[ -5 -3 ] [ -5 ]
Calculating the values:
X = [ (-1/10) * ( -95 + 25 ) ]
[ (-1/10) * ( -95 ) ]
Simplifying:
X = [ (-1/10) * ( -70 ) ]
[ 9.5 ]
Therefore, the solution to the given system of equations is x = -7 and y = 9.5.
The correct ordered pair is (-7, 9.5).