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Functions and Equations of Two Variables

Functions and Equations of Two Variables. Lesson 6.1. Functions of Two Variables. Consider a function with two inputs and one output Two independent variables One dependant variable z = f ( x, y ). 5. 7. f (x, y). 43. Example. Consider the volume of a cylinder Given r = 5, h = 10

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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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