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# 8.2 Equations W Two Variables

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8.2: Equations with Two Variables. Making tables and graphing ordered pair solutions.

8.2: Equations with Two Variables. Making tables and graphing ordered pair solutions.

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• 1. Chapter 8 Section 2: Equations with Two Variables. February 5 th , 2009
• 2. Warm Up <ul><li>Graph these points : </li></ul><ul><li>A (-1, 4) </li></ul><ul><li>B (4, 2) </li></ul><ul><li>C (-3, -2) </li></ul><ul><li>Evaluate </li></ul><ul><li>3x – 5, for x = 2 </li></ul><ul><li>2x + 3y, for x = -4 and y =5. </li></ul><ul><li>Is this Relation a Function ? </li></ul><ul><li>{(1, 2), (2, 2), (3, 1), (2, 4)} </li></ul>
• 3. Two Variables <ul><li>We’ve talked about equations like this: </li></ul><ul><li>2x + 5 = 7x (SADMEP &amp; Combine Like Terms) </li></ul><ul><li>But what about equations that look like this: </li></ul><ul><li>y = 3x + 4 (???) </li></ul><ul><li>An ORDERED PAIR that will make a two variable statement true is called a Solution to the equation. </li></ul><ul><li>Two variables, x = ? and y = ?, so (?, ?) is the ordered pair that is a solution to a particular two variable equation. </li></ul>
• 4. Example <ul><li>Some times we’ll just give you one of the numbers in the ordered pair. </li></ul><ul><li>You plug it in, and chug out the other variable in the ordered pair. </li></ul><ul><li>Like this: y = 3 x + 4, when x = -1 . </li></ul><ul><li>y = 3 (-1) + 4 </li></ul><ul><li> y = -3 + 4 </li></ul><ul><li>y = 1 </li></ul><ul><li>The ordered pair is then, ( -1 , 1), for this equation. </li></ul>
• 5. Try These: Find y, and give Ordered Pair, x = -3 <ul><li>y = 2x + 1 </li></ul><ul><li>y = -4x + 3 </li></ul><ul><li>y = 0x -4 </li></ul>y = -5, (-3, -5) y = -15, (-3, 15) y = -4, (-3, -4)
• 6. Real Life Example <ul><li>The equation t = 21 – 0.01n models the normal low July temperature in degrees Celsius at Mt. Rushmore, South Dakota. In the equation, t is the temperature at n meters above the base of the mountain. Find the normal low July temperature at 300m above the base. </li></ul>t = 21 – 0.01(300) t = 21 – 3 t = 18 degrees Celsius The ordered pair solution for this equation is (300, 18). 300 is the domain, 18 is the range.
• 7. Try This One <ul><li>Determine the normal low July temperature at 700m above the base of Mt. Rushmore, using the same equation as before: t = 21 – 0.01n. </li></ul><ul><li>Remember, n = the meters above the base, t = temperature. </li></ul><ul><li>The low temperature in July at Mt. Rushmore is 14 degrees Celsius. </li></ul>
• 8. Graphing Equations with Two Variables. <ul><li>Equations with two variables can have many solutions..Duh. </li></ul><ul><li>Show ALL of these solutions by graphing . </li></ul><ul><li>Any Equation , that when graphed , forms a line , is a LINEAR EQUATION. </li></ul>
• 9. Graph: y = -0.5x + 3 <ul><li>Make a table of values to show ordered-pair solutions. </li></ul><ul><li>Graph the ordered pairs. Draw a line through the points. </li></ul>(4, 1) y = -0.5(4) + 3, y = -2 + 3, y = 1 x = 4 (0, 3) y = -0.5(0) + 3, y = 0 + 3, y = 3 x = 0 (-2, 4) y = -0.5(-2) + 3, y = 1 + 3, y = 4 x = -2 Ordered Pairs (x, y) Equation: y = -0.5x + 3 Values for X (that we make up)
• 10. Graph These For Me In Your Notes, The Connect The Dots <ul><li>y = 2x + 1 </li></ul><ul><li>y = 3x -2 </li></ul>Pick reasonable numbers for x so that you can solve for y and find the ordered pair that allows you to graph the equation. You need three points to graph so that you know you have done it correctly.
• 11. Linear Equations and Vertical Line Test <ul><li>The graph y = -0.5x + 3 passes the Vertical Line Test. </li></ul><ul><li>Therefore, this equation, the RELATION, is a FUNCTION! </li></ul><ul><li>A Linear Equation is a Function UNLESS its graph is a vertical one. </li></ul><ul><li>…Wait, what does a vertical graph look like? </li></ul>
• 12. Graph These Equations <ul><li>Y = 2 </li></ul><ul><li>X = 2 </li></ul>For every value of x, y ALWAYS equals 2. No matter what. For every value of y, x ALWAYS equals 2. No matter what. This is the graph for y = 2, it is a horizontal line. Does it pass the vertical line test? This is the graph for x = 2, it is a vertical line. Does it pass the vertical line test? Yes! Y = 2 is a function! No! X = 2 is NOT a function!
• 13. One More Example <ul><li>Solve for y, then Graph. </li></ul><ul><li>3x + y = -5 </li></ul><ul><li>3x – 3x + y = -5 – 3x (Subtract 3x from both sides) </li></ul><ul><li>y = -3x - 5 (Solved for y) </li></ul><ul><li>Make a Table. </li></ul><ul><li>Graph. </li></ul>(x, y) -3x – 5 X
• 14. Solve each equation for y, then graph. <ul><li>2x + y = 3 </li></ul><ul><li>y – x = 5 </li></ul><ul><li>-3x + 2y = 6 </li></ul><ul><li>Remember, you come up with your own domain (or x value). Try and make it easy on yourself by picking easy numbers (like: -2, -1, 0, 1, and 2). </li></ul>
• 15. Assignment #2 <ul><li>Pages 394-395: 1-35 all. </li></ul><ul><li>YOU MUST GRAPH!!! </li></ul><ul><li>Because of the Field Trip, Due Monday . </li></ul>