The solutions to the following system of equation is x = 0, y = 2, z = 5.
Solving system of equation using Gaussian Elimination
Gaussian Elimination is an approach for solving systems of linear equations that involves reducing the augmented matrix of the system into a smaller, row-equivalent matrix which corresponds to the given system of equations.
Given the system of equation:
5x + 9y - 3z = 3 ------ (1)
-3x - 9y + 4z = 2 ------(2)
4x + 9y - 5z = -7 ------(3)
Multiply first equation by 3/5 and add the result to the second equation, we have:
5x + 9y - 3z = 3
-18/5y + 11/5z = 19/5
4x + 9y - 5z = -7
Multiply first equation by -4/5 and add the result to the third equation, we have:
5x + 9y - 3z = 3
-18/5y + 11/5z = 19/5
9/5y - 13/5z = -47/5
Remove the fractions by multiplying second and the third equation by 5, we have:
5x + 9y - 3z = 3
-18y + 11z = 19
9y - 13z = -47
Multiply second equation by 1/2 and add the result to the third equation, we have:
5x + 9y - 3z = 3
-18y + 11z = 19
-15/2z = -75/2
Remove the fractions by multiplying third equation by 2, we have;
5x + 9y - 3z = 3
-18y + 11z = 19
-15z = -75
Divide the third equation by 15, we have:
5x + 9y - 3z = 3
-18y + 11z = 19
-z = -5
Solve for z
z = 5
From the second equation, let's solve for y:
-18y + 11z = 19
-18y + (11×5) = 19
-18y + 55 = 19
-18y = 19 - 55
-18y = -36
y = 2
Replace the value of y and z into the first equation to solve for x;
5x + 9y - 3z = 3
5x + 9(2) - 3(5) = 3
5x + 3 = 3
5x = 0
x = 0/5
x = 0
Therefore, the solutions to the following system of equation is x = 0, y = 2, z = 5.