In an equation with two variables x and y , the graph is a line. In an equation with three variables x , y and z the graph is a plane. Think of a plane as a wall that extends vertically, horizontally or diagonally indefinitely. The techniques used to solve these systems are basically the same. The first thing you want to do is rewrite all equations in Ax + By + Cz = D form where A , B , C and D are numbers. Choose any two equations and eliminate a variable. Choose two other equations and eliminate the same variable. The idea is to reduce the original system into two equations with two variables. Then you can solve the system as you would a system with two variables and two equations. The solution is the intersection of the planes. A few examples will clarify this process.
Example: Solve the system x + 2y + 3z = 14
2x + y – 2z = -2
3x – 4y + z = -2
First we choose a variable we wish to eliminate. Let’s eliminate the x variable by using the first and second equation. x + 2y + 3z = 14
2x + y – 2z = -2
Multiply the first equation by -2 to get
-2x – 4y – 6z = -28
2x + y – 2z = -2
When adding the equations the x terms are eliminated and we have left -3y – 8z = -30.
Now choose two other equations and eliminate the x terms. Let’s use the first equation and the third equation. The reason why we are using the first and third equations is because we only have to multiply the first equation before adding to eliminate the x term. Using the second and third equations would require multiplying both equations or multiplying one or the other equation by a fraction, which is more difficult.
Multiply the first equation by -3 to get
-3x – 6y -9z = -42
3x – 4y + z = -2
Add the equations to get -10y – 8z = -44.
Now we have a system of equations in two variables.
-3y – 8z = -30
-10y – 8z = -44
Notice the z terms are the same, so we can subtract the equations to eliminate the z terms. This gives us 7y = 14, therefore y = 2.
Substitute 2 in for y in either equation to get -3(2) – 8z = -30 and z = 3. To solve for x, use one of the original equations with y = 2 and z = 3. Using the first equation we get x + 2(2) + 3(3) = 14. Therefore, x = 1. The solution to the equation is x = 1, y = 2, z = 3.
Sometimes a system of equations will be dependent. This occurs when any two of the original equations are the same. Systems can also be inconsistent. This occurs when the coefficients of the variables in two equations are the same but the constants are different. For example, if two of the equations are 2x + 3y + 4z = 1 and 2x + 3y + 4z = -2, the systems are inconsistent. The same equations cannot equal both 1 and -2.
This guide and examples should ease any confusion about systems of linear equations with three variables.
