System of Equations (3 Variables)

How to solve the system of equations with 3 variables: 2 examples and their solutions.

Example 1

Example

Solution

Choose the target variable to remove.

Choose z as the variable to remove.

Elimination Method

Choose two equations
and make the z terms the same or the opposite.

First choose the first and the second equations.

The z terms are already the same: +z.

So write 2x - y + z = 6 and x + y + z = 3.

To remove the z terms,
subtract the equations.

2x - x = x
-y - y = -2y
The z terms are cancelled.
6 - 3 = 3

So x - 2y = 3.

Choose another pair of equations
and make the z terms the same or the opposites.

Choose the first and the third equations.

The z terms are already the opposites: +z and -z.

So write x + y + z = 3 and x + 2y - z = -4.

To remove the z terms,
add the equations.

x + x = 2x
+y + 2y = +3y
The z terms are cancelled.
3 + (-4) = -1

So 2x + 3y = -1.

So, by removing the variable z,
you get these two equations:
x - 2y = 3 and 2x + 3y = -1.

Choose the target variable to remove.

Choose x as the variable to remove.

Make the x terms the same or the opposites.

To change x to 2x,
multiply 2 to both sides of x - 2y = 3.
Then 2x - 4y = 6.

And write 2x + 3y = -1.

To remove the x terms,
subtract the equations.

The x terms are removed.
-4y - 3y = -7y
6 - (-1) = 6 + 1 = 7

So -7y = 7.

Divide both sides by -7.
Then y = -1.

Put y = -1 into either x - 2y = 3 or 2x + 3y = -1.

x - 2y = 3 is simplier.

So put y = -1 into x - 2y = 3.
Then x - 2⋅(-1) = 3.

Substitution Method

Solve x - 2⋅(-1) = 3.
Then x = 1.

Put x = 1 and y = -1 into one of the given equations.

x + y + z = 3 is simpler.

So put x = 1 and y = -1 into x + y + z = 3.

Then 1 + (-1) + z = 3.

1 + (-1) + z = 3
Cancel 1 and -1.
Then z = 3.

So x = 1, y = -1, z = 3 is the answer.

Example 2

Example

Solution

Just like the previous example,
you can solve this system by finding x, y, and z.

But, in some cases,
finding the pattern might save your time.

See the given equations.
The coefficients of x, y, and z are all the same:
1 and 4.

And it says to find x + y + z,
not to find x, y, and z.

So, to make x + y + z,
add the given equations.

x + 4x = 5x
y + 4y = 5y
4z + z = 5z
5 + 7 + 8 = 12 + 8 = 20

So 5x + 5y + 5z = 20.

5x + 5y + 5z = 5(x + y + z)

Common Monomial Factor

Divide both sides by 5.
Then x + y + z = 4.

So 4 is the answer.