Solve Systems of Linear Equations in Three Variables Chapter 3.4

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

• We still have the x- and y-axes, but we add the z-axis

• A point is three dimensions is an ordered triple:

• The next slide shows the point , which is:• 1 unit from the origin in the x-direction• Then 1 unit from here in the y-direction• Finally, 1 unit from here in the z-direction

Three Dimensional Space

Linear Equations in 3-Space

• A linear equation in three dimensional space (or 3-space for short) has the following form: , where a, b, and c are not all zero

• Although these are called “linear equations”, in 3-space they form, not lines, but planes

• What makes them “linear” is that the variables x, y, and z have exponent 1

• The next slide shows the linear equation

Linear Equations in 3-Space

Linear Equations in 3-Space

• A system of linear equations in 3 variables takes the form

• Each of the a, b, and c with subscripts represent different coefficients (usually as integers)

• If a system has a solution, it is the ordered triple that holds true for each equation

• The solution is the point at which the three planes intersect

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

• The solution to a system of linear equations in 3-space is an ordered triple that satisfies all three equations

• Example: Is the point is a solution to the system below?

Solutions

• The solution to a system of linear equations in 3-space is an ordered triple that satisfies all three equations

• Example: Is the point is a solution to the system below?

• The point is a solution because

Guided Practice

Determine whether the given point is the solution to the system of equations.

a) for

b) for

Guided Practice

Determine whether the given point is the solution to the system of equations.

a) for , not a solution

b) for , is a solution

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: Solve the system of equations in 3 variables

Solving

• To begin, you must decide on a variable to eliminate

• You can eliminate any of the three variables, but try to choose the variable that seems easiest to eliminate

• For example, since the coefficient of the y term in the third equation is , it might be easiest to eliminate the y terms

• This will be done in two steps

• First pick two of the three equations and eliminate y terms

• Then pick the remaining equation and one of the first two equations and eliminate x terms

• You will be left with two equations containing x and z

Solving

Example: Solve the system of equations in 3 variables

For which two equations will you choose to eliminate the y term?

Solving

Example: Solve the system of equations in 3 variables

For which two equations will you choose to eliminate the x term?

It would probably be easiest to with the first and third equations (why?)

Solving

• Remember that the y terms must cancel to zero

• If we pick the first and third equations, what should we do so that the y terms cancel?

Solving

• Remember that the y terms must cancel to zero

• If we pick the first and third equations, what should we do so that the x terms cancel?

• We will need to multiply the third equation by

Solving

• Now, we put the resulting equation aside and eliminate y for the second and third equations

Solving

• We now have two equations with only x and z terms

• We must choose a variable to eliminate

• Since the x terms cancel easily, we add the two equations

Solving

Now, find x by substituting the value for z into either of the above equations

Solving

• Finally, substitute the values and into any of the original 3 equations and solve for y

• I choose the third equation

• The solution is

Example

Solve the system