Graphing Linear Equations in Three Variables GOAL Graph linear equations in three variables. Key Words • three-dimensional coordinate system • z-axis • ordered triple • octants • linear equation in three variables

Points in a two-dimensional coordinate system are represented by ordered pairs. Each point in space can be represented by an ordered triple (x, y, z), such as (1, -2, 3). The three axes, taken two at a time, determine three coordinate planes. These planes divide space into eight octants. The first octant is the one for which all three coordinates are positive.

Plot the ordered triple in a three-dimensional coordinate system. Example 1 Plot Points in Three Dimensions Plot the ordered triple in a three-dimensional coordinate system. a. 5, 3, 4 ( ) – b. 3, 4, 2 ( ) – a. 5, 3, 4 ( ), – 5, 3 ) SOLUTION To plot first find the point in the xy-plane. The point lies 4 units above it.

Example 1 Plot Points in Three Dimensions b. To plot find the point in the xy-plane. The point lies 2 units below it. 3, 4, 2 ( ) – 3, 4 4

Plot the ordered triple in a three-dimensional coordinate system. Checkpoint Plot Points in Three Dimensions Plot the ordered triple in a three-dimensional coordinate system. 1. 1, 2, 4 ( ) ANSWER

Plot the ordered triple in a three-dimensional coordinate system. Checkpoint Plot Points in Three Dimensions Plot the ordered triple in a three-dimensional coordinate system. 2. 2, 1, 3 ( ) – ANSWER

Plot the ordered triple in a three-dimensional coordinate system. Checkpoint Plot Points in Three Dimensions Plot the ordered triple in a three-dimensional coordinate system. 3. – – – ( 4, 1, 5 ) ANSWER

Plot the ordered triple in a three-dimensional coordinate system. Checkpoint Plot Points in Three Dimensions Plot the ordered triple in a three-dimensional coordinate system. 4. 3, 2, 4 ( ) – ANSWER

The graph of a linear equation in three variables is a plane. A linear equation in three variables x, y, and z is an equation of the form ax + by + cz = d where the constants a, b, and c are not all zero. An ordered triple (x, y, z) is a solution of a linear equation in three variables if the values of x, y, and z make the equation true. The graph of a linear equation in three variables is a plane. In a three-dimensional coordinate system, the graphs of equations in one or two variables are also planes.

Find the points where the graph intersects the axes. Example 2 Graph a Linear Equation in Three Variables Sketch the graph of 3x 2y 4z 12. Label the points where the graph crosses the x-, y-, and z-axes. + = Find the points where the graph intersects the axes. First let x 0 and y 0, and then solve for z. SOLUTION = Substitute 0 for x and for y. 12 = + 4z 3 ( ) 2 Simplify. 12 = 4z 3 = z Solve for z. The z-intercept is 3, so plot the point . ( ) 0, 0, 3

Next, let y 0 and z 0, and then solve for x. = = Example 2 Graph a Linear Equation in Three Variables Next, let y 0 and z 0, and then solve for x. = = Substitute 0 for y and for z. 12 = + 3x 2 ( ) 4 Simplify. 12 = 3x 4 = x Solve for x. The x-intercept is 4, so plot the point . Finally, let x 0 and z 0, and then solve for y. = ( ) 4, 0, 0 Substitute 0 for x and for z. 12 = + 2y 3 ( ) 4 Simplify. 12 = 2y 6 = y Solve for y. The y-intercept is 6, so plot . ( ) 0, 6, 0

Example 2 Graph a Linear Equation in Three Variables Connect the points with lines. The lines form the triangular region of the plane that lies in the first octant.

Checkpoint Graph a Linear Equation in Three Variables Sketch the graph of the equation. Label the points where the graph crosses the x-, y-, and z-axes. 5. 6 = + x 3y 2z ANSWER

Checkpoint Graph a Linear Equation in Three Variables Sketch the graph of the equation. Label the points where the graph crosses the x-, y-, and z-axes. 6. 10 = + y 5x 2z ANSWER

Checkpoint Graph a Linear Equation in Three Variables Sketch the graph of the equation. Label the points where the graph crosses the x-, y-, and z-axes. 7. 4 = y + x ANSWER

GARDENING Tulips come in a variety of sizes, shapes, and colors GARDENING Tulips come in a variety of sizes, shapes, and colors. The bulbs must be planted in late fall or early winter in order to bloom the following spring.

Write a linear model for the total cost of the bulbs Example 3 Model a Real-Life Situation Gardening You are planting tulip bulbs and daffodil bulbs in a community garden. Tulip bulbs cost $6 per dozen, and daffodil bulbs cost $5 per dozen. Fertilizer for the bulbs costs $10. Write a linear model for the total cost of the bulbs and the fertilizer for the garden. b. Evaluate the model for several different numbers of tulip bulbs and daffodil bulbs. Organize your results in a table. 18

Your total cost involves two variable costs (for the Example 3 Model a Real-Life Situation SOLUTION Your total cost involves two variable costs (for the two types of bulbs) and one fixed cost (for the fertilizer). VERBAL MODEL Total cost • = Dozens of tulip bulbs Cost of (per dozen) + daffodil Cost fertilizer 19

Cost of tulip bulbs (per dozen) 6 (dollars) = Example 3 Model a Real-Life Situation LABELS Total cost C (dollars) = Cost of tulip bulbs (per dozen) 6 (dollars) = Dozens of tulip bulbs t (dozens of bulbs) = Cost of daffodil bulbs (per dozen) 5 (dollars) = Dozens of daffodil bulbs d (dozens of bulbs) = Cost of fertilizer 10 (dollars) = ALGEBRAIC MODEL = C 6t + 5d 10 20

t (dozens of tulip bulbs) Example 3 Model a Real-Life Situation b. To evaluate the model, substitute values for t and d. For example, substitute 4 for t and 3 for d. = C 6t + 5d 10 Write algebraic model. = C 6 + 5 Substitute 4 for t and 3 for d. ( ) 4 3 10 = C 49 Simplify. The table shows the total cost for several more values of t and d. t (dozens of tulip bulbs) daffodil bulbs) d (dozens of 2 4 6 1 3 $27 $32 $37 $39 $44 $49 $51 $56 $61 The total cost is $49 for 4 dozen tulip bulbs and 3 dozen daffodil bulbs.

Use the given coordinates to find the coordinates of the vertices J, K, L, and M of the rectangular prism.

Use the given point to find the volume of the rectangular prism.

Geometry Skills: Parallel Lines and Angles Find the values of x and y.

Slip

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