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5.2 GRAPHING THREE VARIABLES An equation such as Ax By Cz D such that A, B, C, and D are not all zero is called a linear equation in three variables. We can regard an equation in one or two variables as an equation in three variables. Example: 4x 3z 5 can be thought of as 4x 0y 3z 5 In space, the graph of a linear equation in three variables is a plane . In order to graph a linear equation in three variables we find the x-, y-, and z-intercepts. a. To find the x-intercept, substitute and solve for x. b. To find the y-intercept, substitute and solve for y. c. To find the z-intercept, substitute and solve for z. zero for both y and z zero for both x and z zero for both x and y We can then plot the intercepts and graph the plane that goes through them. 1. Sketch the graph of the equation 2x 3y 6z 12 x-intercept: y-intercept: z-intercept: 2. Sketch the graph of the equation 5x 2z 10 x-intercept: y-intercept: z-intercept: What did you find about the graph? The plane is parallel to the y-axis If the coefficient of a variable in an equation of a zero plane is , then the plane is parallel to the axis of that variable if the constant term is not zero. 3. Sketch the graph of the equation 3x 2z 0 x-intercept: y-intercept: z-intercept: What did you find about the graph? The plane contains the y-axis If the coefficient of a variable in an equation of a zero plane is , then the plane contains the axis of that variable if zero the constant term is . 4. Sketch the graph of the equation y3 x-intercept: y-intercept: z-intercept: What did you find about the graph? The plane is parallel to the x-axis and z-axis The plane is parallel to the xz-plane two If the coefficients of of the variables are zero, but the constant term is not zero, then the parallel graph is to the other two axes parallel and to the plane of the other two variables. Questions 5-8: A. Name the x-, y-, and z-intercepts. B. Determine whether or not the graph is parallel to or contains any of the coordinate axes, and if so, which one(s). C. Determine whether or not the graph is parallel to or coincides with one of the coordinate planes, and if so, which one. D. Graph. 5. 6x 3y 2z 18 6. 3y 2z 6 7. x 2 8. 3x 5y 3