Section 3.5 - Graphing Linear Equations in Three Variables ALGEBRA TWO CHAPTER THREE: SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES Section 3.5 - Graphing Linear Equations in Three Variables

LEARNING GOALS Goal One - Graph linear equations in three variables and evaluate linear functions of two variables. Goal Two - Use functions of two variables to model real-life situations.

VOCABULARY A linear equation in three variables, x, y, and z is an equation of the form ax + by + cz = d, where a, b, and c are all not zero. In a three-dimensional coordinate system, the xy-plane is in a horizontal position, with the z-axis as a vertical line through the origin. 1

VOCABULARY An ordered triple is given by (x, y, z) and represents a point in space. 1

Graphing a 3 variable equation STEP 1a: Find the points where the graph intersects the axes. Let y = 0 and z = 0, and solve for x. STEP 1b: Find the points where the graph intersects the axes. Let x = 0 and z = 0, and solve for y. STEP 1c: Find the points where the graph intersects the axes. Let x = 0 and y = 0, and solve for z. STEP 2: Plot the intercepts on an xyz- coordinate plane. STEP 3: Draw lines to connect the points. STEP 4: Shade in the triangle formed by the lines.

Sketch the graph of -2x + 4y + 8z = 16 Graphing a Linear Equation in Three Variables Sketch the graph of -2x + 4y + 8z = 16 STEP 1a: Find the points where the graph intersects the axes. Let y = 0 and z = 0, and solve for x. -2x + 4y + 8z = 16 The x-intercept is the point (-8, 0, 0) -2x + 4(0) + 8(0) = 16 -2x/-2 = 16/-2 x = -8 1

Sketch the graph of -2x + 4y + 8z = 16 Graphing a Linear Equation in Three Variables Sketch the graph of -2x + 4y + 8z = 16 STEP 1b: Find the points where the graph intersects the axes. Let x = 0 and z = 0, and solve for y. -2x + 4y + 8z = 16 The y-intercept is the point (0, 4, 0) -2(0) + 4y + 8(0) = 16 4y/4 = 16/4 y = 4 1

Sketch the graph of -2x + 4y + 8z = 16 Graphing a Linear Equation in Three Variables Sketch the graph of -2x + 4y + 8z = 16 STEP 1c: Find the points where the graph intersects the axes. Let x = 0 and y = 0, and solve for z. -2x + 4y + 8z = 16 The z-intercept is the point (0, 0, 2) -2(0) + 4(0) + 8z = 16 8z/8 = 16/8 z = 2 1

Sketch the graph of -2x + 4y + 8z = 16 Graphing a Linear Equation in Three Variables Sketch the graph of -2x + 4y + 8z = 16 STEP 2: Plot the intercepts on an xyz- coordinate plane. The x-intercept is the point (-8, 0, 0) The y-intercept is the point (0, 4, 0) The z-intercept is the point (0, 0, 2) 1

Sketch the graph of -2x + 4y + 8z = 16 Graphing a Linear Equation in Three Variables Sketch the graph of -2x + 4y + 8z = 16 STEP 3: Draw lines to connect the points. 1

Sketch the graph of -2x + 4y + 8z = 16 Graphing a Linear Equation in Three Variables Sketch the graph of -2x + 4y + 8z = 16 STEP 4: Shade in the triangle formed by the lines. 1

Evaluate a function in two variables Step 1: Solve the equation for z. Step 2: Substitute the given values of x and y Step 3: Simplify to find z ( or f(x,y) )

Evaluating a Function of Two Variables (1) Solve the equation for z. Write the linear function 10x - 3y + 12z = - 60 as a function of x and y. Evaluate the function for f (3, -2) (1) Solve the equation for z. 10x - 3y + 12z = -60 Write original equation 12z = -10x + 3y + -60 Bring 10x and 3y to right side. z = (-5/6)x + (1/4)y - 5 Divide everything by 12 f (x, y) = (-5/6)x + (1/4)y - 5 Replace z with f (x, y) 1

Evaluating a Function of Two Variables Write the linear function 10x - 3y + 12z = - 60 as a function of x and y. Evaluate the function for f (3, -2) (2) Substitute 3 for x and -2 for y. f (3, -2) = (-5/6)(3) + (1/4)(-2) - 5 Substitute f (3, -2) = (-5/2) + (-1/2) - 5 Multiply f (3, -2) = (-6/2) - 5 Add fractions The graph contains the point (3, -2, -8) Simplify f (3, -2) = (-3) - 5 f (3, -2) = -8 Simplify 1

ASSIGNMENT READ & STUDY: pg. 170-172. WRITE: pg. 173-175. #27, #31, #37, #39, #45, #51