Solve Systems of Linear Equations in Three Variables Chapter 3.4

Three Dimensional Space A number line occupies one dimension because it has only one kind of direction: left-right The coordinate plane uses two number lines set at right angles to each other and occupies two dimensions because it has two kinds of direction: left-right and up-down We can extend this idea to three dimensions by adding a third number line that is mutually perpendicular to the other two In three dimensions, the three number lines occupy three dimensions and there are three kinds of direction: left-right, up-down, and backward-forward

Three Dimensional Space In one dimension, a position on the number line can be specified with a single number In two dimensions (the coordinate plane), we need two numbers: one to specify the left-right distance from zero and another to specify the up-down distance from zero In three dimensions, we need three numbers: One to specify the left-right distance from zero One to specify the up-down distance from zero One to specify the backward-forward distance from zero

Three Dimensional Space

Linear Equations in 3-Space

As with a linear system in 2 dimensions, a system in 3-space may have: Exactly one solution, or An infinite number of solutions, or No solution The next slide (the image taken from your textbook) shows how the planes might look in each of the three situations

Linear Equations in 3-Space

Solutions

Guided Practice

Solving Solving a system of linear equations in 3-space is like solving a system in 2-space (in the plane) We can solve by substituting or by elimination Of these two methods, the elimination method is easier, so we will only use this method However, finding the solution is a bit longer and requires that you organize your work and concentrate on what you are doing!

Solving

Example