A System of three linear equations looks like this:
x + 2y - 3z = -3 Equation 1 2x - 5y + 4z = 13 Equation 2 5x + 4y - z = 5 Equation 3
It is just three linear equations put together. There will always be three variables in a system of 3 linear equations. In this case, they are x, y, and z.
Solving a system of three equations by the linear combination method only takes 3 steps.
Step 1: Rewrite the equations in the system so that it becomes a linear system with two variables.
Step 2: Solve the new linear system for both of its variables
Step 3: Substitute the values of the variables found in Step 2 in to one of the original equations and solve for last variable.
Note: If you get 0=1 as one of the equations, then the system has no solution. If You get 0=0, then there are infinitely many solutions.
Solving a system with Linear Combination Method:
3x + 2y + 4z = 11 2x - y + 3z = 4 5x - 3y + 5z = -1Step 1: Eliminate one of the variables in two of the original equations.
Add 2 times the second equation to the first.
3x + 2y + 4z = 11 Equation 1
4x - 2y + 6z = 8 Equation 2 times 2
7x + 10z= 19 New Equation 1
Add -3 times the second equation to the third.
5x - 3y + 5z = -1 Equation 3
-6x + 3y - 9z = -12 Equation 2 times -3
-x - 4z = -13 New Equation 2
Step 2: Solve the new system of linear equations in two variables.
7x + 10z = 19 New Equation 1
-7x - 28z = -91 New Equation 2 times 7
-18z = -72
z = 4 Solve for z
x = -3 Substitute 4 for z into New Equation 1 or 2 to find x
Step 3: Substitute x = -3 and z = 4 into an original equation and solve for y.
2x - y + 3z = 4 Equation 2
2(-3) - y + 3(4) = 4 Substitute -3 for x and 4 for z
y = 2 Solve for y
Now you know x = -3, y = 2, and z = 4. You can also write it as (-3,2,4). To check your answer, substitute the values for x back in to each of the original equations.
Example Problems:
1.
x + 2y + 5z = -1 2x - y + z = 2 3x + 4y - 4z = 14
2.
5x - 4y + 4z = 18 -x+ 3y - 2z = 0 4x - 2y + 7z = 3
3.
-5x + 3y + z = -15 10x + 2y + 8z = 18 15x + 5y + 7z = 9