We can rewrite the system of equations as a matrix equation:
[ 1.5 -0.5 ] [ x ] = [ -3.5 ]
[ 1.5 1.5 ] [ y ] [ -7.5 ]
We can represent this matrix equation as AX = B, where A is the matrix of coefficients, X is the column vector of variables (x and y), and B is the column vector of constants.
To solve for X, we can multiply both sides of the equation by the inverse of matrix A:
A^(-1) * AX = A^(-1) * B
Since A is a 2x2 matrix, we can easily find its inverse:
[ 1.5 -0.5 ] [ x ] = [ -3.5 ]
[ 1.5 1.5 ] [ y ] [ -7.5 ]
The inverse of A is:
[ 1/2 1/2 ]
[ -1/2 3/2 ]
Multiplying both sides by the inverse of A gives us:
[ 1/2 1/2 ] [ 1.5x−0.5y ] = [ 1/2 * -3.5 + 1/2 * -7.5 ]
[ -1/2 3/2 ] [ 1.5x+1.5y ] [ -1/2 * -3.5 + 3/2 * -7.5 ]
Simplifying the right side of the equation gives us:
[ 1/2 1/2 ] [ 1.5x−0.5y ] = [ -1.75 ]
[ -1/2 3/2 ] [ 1.5x+1.5y ] [ -7 ]
We can now solve for X:
[ x ] = [ 1/2 1/2 ]^-1 * [ -1.75 ]
[ y ] [ -1/2 3/2 ] [ -7 ]
Using the inverse of matrix [ 1/2 1/2 ], we have:
[ x ] = [ -2 1 ] * [ -1.75 ]
[ y ] [ 1 -1] [ -7 ]
This simplifies to:
[ x ] = (2 * -1.75) + (1 * -7)
[ y ] (-2 * -1.75) + (-1 * -7)
Calculating the values:
[ x ] = -3.5 - 7
[ y ] 3.5 - 7
Simplifying further:
[ x ] = -10.5
[ y ] -3.5
Therefore, the solution to the system of equations is x = -10.5 and y = -3.5.
Solve the system of equations using a matrix
1.5x−0.5y=−3.5
1.5x+1.5y=-7.5
1 answer