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Functions and Equations of Two Variables

This lesson explores functions with two independent variables and one dependent variable, focusing on solving equations graphically and symbolically. It includes practical examples, such as calculating the volume of a cylinder, and guides students through solving systems of linear and non-linear equations. Techniques like substitution and graphing are emphasized for finding solutions, alongside using calculators for efficiency. By the end, students should grasp how to handle equations in two variables and understand the nature of solutions in different scenarios.

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Functions and Equations of Two Variables

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  1. Functions and Equations of Two Variables Lesson 6.1

  2. Functions of Two Variables • Consider a function with two inputs and one output • Two independentvariables • One dependant variable • z = f ( x, y ) 5 7 f (x, y) 43

  3. Example • Consider the volume of a cylinder • Given r = 5, h = 10 • V = π *25*10 = 250π • Calculator can define such functions r h

  4. Solving for One of the Variables • How high must the cylinder be for • Radius of 6 inches • Volume of 230 cubic inches • Write out the formula • Substitute in the known quantities • Solve for the unknown value 6 h 230 in3

  5. Linear Equation in Two Variables • Format • Where a, b, and k are constants • This can also be thought of as a function in two variables • Example • Now note that there are many (x, y) ordered pairs that can be considered solutions

  6. System of Equations • If we have two equations in two variables it is possible that one ordered pair is the solution for both equations • Which of the following ordered pairs are solutions for the system? (3, 2) (3, -4) (5, 0)

  7. Solving Systems of Equations • Graphical solution • Solve each equation for y • Graph the resulting function • Note their intersections

  8. Symbolic Solution • Solve one of the equations for one of the variables • x = y + 5 • Substitute the expression in for that variable in the other equation • 2 ( y + 5 ) + y = 10 • Result is an equation in one variable • Solve that equation for the variable 3y = 0 • Substitute that value back into the other equation 2x + 0 = 10

  9. Try It Out • Given • Part of class determine graphical solution • Part of class determine symbolic solution by substitution

  10. Using Calculator • Calculator can be used to solve systems of equations • Use solve command • Note use of • and • curly brackets { }

  11. Systems of Non-Linear Equations • Consider • Note one of equations is not linear • Same types of solutions can often be used • Symbolic by substitution • Graphical • What kind of graphs are demonstrated?

  12. Number of Solutions • System of linear equations • One solution no solutions many solutions • For non linear systems • Depends on the type of the graphs involved • What different possibilities exist for a line and a parabola?

  13. Try It Out • Given the system • Part of class do graphically • Part of class do symbolically

  14. Assignment • Lesson 6.1 • Page 460 • Exercises 1 – 75 EOO

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