Solve systems of equations with three variables using addition/elimination

Solving systems of equations with 3 variables is very similar to how we solve systems with two variables. When we had two variables we reduced the system down to one with only one variable (by substitution or addition). With three variables we will reduce the system down to one with two variables (usually by addition), which we can then solve by either addition or substitution.

Also Read : Solve systems of equations using substitution

To reduce from three variables down to two it is very important to keep the work organized. We will use addition with two equations to eliminate one variable. This new equation we will call (A). Then we will use a different pair of equations and use addition to eliminate the same variable. This second new equation we will call (B). Once we have done this we will have two equations (A) and (B) with the same two variables that we can solve using either method. This is shown in the following examples.

Also Read:

Also Read : Solve systems of equations using the addition/elimination method

Example :


3x + 2y – z = -1

-2x – 2y + 3z = 5

5x + 2y – z = 3


We will eliminate y using two different pairs of equations.

Using the first two equations,

Add the first two equations

This is equation (A), our first equation

Using the second two equations. Add the second two equations. This is equation (B), our second equation

Using (A) and (B) we will solve this system.

(A).x + 2z = 4

(B).3x + 2z = 8

We will solve by addition

Multiply (A) by -1

-1(x +2z) =(4)( -1)

-x – 2z = -4

Add to the second equation, unchanged

x = 2

We now have x! Plug this into either (A) or (B)

We plug it into (A), solve this equation, subtract 2

(2) + 2z =4

2z = 2

z = 1

We now have z! Plug this and x into any original equation

We use the first, multiply 3(2) = 6 and combine with -1

3(2) + 2y -(1) = -1

6 + 2y – 1 = -1

2y + 5 = -1

Solve, subtract 5

2y + 5 – 5 = -1 – 5

2y = -6

y = -3

our solution is ( 2, -3, 1).

As we are solving for x, y, and z we will have an ordered triplet (x, y, z) instead of just the ordered pair (x, y). In this above problem, y was easily eliminated using the addition method. However, sometimes we may have to do a bit of work to get a variable to eliminate. Just as with addition of two equations, we may have to multiply equations by something on both sides to get the opposites we want so a variable eliminates. As we do this remember it is important to eliminate the same variable both times using two different pairs of equations.

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