1. Solving Systems of Equations

2. Graphing There are three methods to solving systems of equations by graphing: 1)Write both equations in slope – intercept form and graph 2)Write both equations in slope-intercept form and graph using the calculator 3)Solve for the x and y intercepts of each equation

3. Graphing Solve the following system of equations by graphing: -6x +3y = -15 -4x +y = -11

4. Graphing – Method 1 Write both equations in slope – intercept form and graph. To do this, solve each equation for y -6x +3y = -15 -4x +y = -11

5. Graphing – Method 1 Writing -6x +3y = -15 in slope intercept form: -6x +3y = -15 +6x 3y = 6x – 15 3 3 3 y = 2x - 5

6. Graphing – Method 1 Writing -4x + y = -11 in slope intercept form: -4x + y = -11 +4x y = 4x - 11

7. Graphing – Method 1 Graph both equations using their slope and y –intercept by starting at the y-intercept and using their slope to do rise over run. Equation 1: y = 2x - 5 y intercept is (0, -5) slope is rise 2, run 1 Equation 2: y = 4x – 11 y intercept is (0, -11) slope is rise 4, run 1

8. Graphing – Method 1 The lines intersect at the point (1,3)

9. Graphing – Method 2 Write both equations in slope-intercept form and graph using the calculator -6x +3y = -15 -4x +y = -11 The equations were already solved for slope-intercept form in method 1, so: y = 2x – 5 y = 4x – 11

10. Graphing – Method 2 (TI-84+) 1)Turn the calculator on 2)Hit the “Y=” key 3)Type in the first equation next to Y 1 4)Use the “X,T,O,n” key to type “X” 5)Hit “Enter” 6)Type in the second equation next to Y 2 7)Hit “Enter”

11. Graphing – Method 2 (TI-84+) 8) Hit the graph button to see the graph

12. Graphing – Method 2 (TI-84+) 9) If necessary, adjust the graph by changing the zoom You can zoom in, or out by hitting the zoom button and then selecting option 2 or 3. Once selected, press enter again when you see the graph Zoom standard goes back to the regular zoom

13. Graphing – Method 2 (TI-84+) 10) When looking at the graph hit the “CALC” button. Do this by hitting the “2ND” key followed by the “TRACE” key 11) Move down to choice five and select “intersect” 12) Press “Enter” and the calculator will return to the graph.

14. Graphing – Method 2 (TI-84+) 13) The calculator will prompt you to select the first curve. Use the arrows to put the blinking cursor on one of the lines 14) Hit “Enter”

15. Graphing – Method 2 (TI-84+) 15) The calculator will prompt you to select the second curve. Use the arrows to put the blinking cursor on the other line (the calculator should have already done this for you) 16) Hit “Enter”

16. Graphing – Method 2 (TI-84+) 17) The calculator will prompt you to guess the location of the intersection. Use the arrow keys to move the flashing curser close to the intersection 18) Hit “Enter”

17. Graphing – Method 2 (TI-84+) 19) The calculator will then tell you the intersection. In this case, “X=3, Y=1” 20) Write your answer as an ordered pair (3,1)

18. Graphing – Method 2 (TI-89) 1)Turn the calculator on 2)Hit the “Y=” key by hitting Diamond + F1 3)Type in the first equation next to Y 1 4)Hit “Enter” 5)Type in the second equation next to Y 2 6)Hit “Enter”

19. Graphing – Method 2 (TI-89) 7) Hit the graph button to see the graph - Do this by hitting diamond and then F3

20. Graphing – Method 2 (TI-89) 8) If necessary, adjust the graph by changing the zoom You can zoom in, or out by hitting the zoom button (F2) and then selecting option 2 or 3. Once selected, press enter again when you see the graph Zoom standard (option 6) goes back to the regular zoom

21. Graphing – Method 2 (TI-89) 9) When looking at the graph select the “Math” menu. Do this by hitting the “F5” key 10) Move down to choice five and select “intersection” 11) Press “Enter” and the calculator will return to the graph.

22. Graphing – Method 2 (TI-89) 12) The calculator will prompt you to select the first curve. Use the arrows to put the blinking cursor on one of the lines 13) Hit “Enter”

23. Graphing – Method 2 (TI-89) 14) The calculator will prompt you to select the second curve. Use the arrows to put the blinking cursor on the other line (the calculator should have already done this for you) 15) Hit “Enter”

24. Graphing – Method 2 (TI-89) 16) The calculator will prompt you to select the lower bound of the intersection. Use the arrow keys to move below or to the left of the intersection 17) Hit “Enter”

25. Graphing – Method 2 (TI-89) 18) The calculator will prompt you to select the upper bound of the intersection. Use the arrow keys to move above or to the right of the intersection 19) Hit Enter

26. Graphing – Method 2 (TI-89) 20) The calculator will then tell you the intersection. In this case, “X=3, Y=1” 21) Write your answer as an ordered pair (3,1)

27. Graphing – Method 3 Graph by solving for the x and y intercepts of each equation: -6x +3y = -9 -4x +y = -8

28. Graphing – Method 3 Find the x and y intercepts of the first equation: -6x +3y = -15 x-intercept, let y=0 -6x +3y = -15 -6x +3(0) = -15 -6x = -15 -6 x = -15/-6 = 5/2 x-int = (5/2,0) y-intercept, let x=0 -6x +3y = -15 -6(0) + 3y = -15 3y = -15 3 y = -5 y-int = (0,-5)

29. Graphing – Method 3 Find the x and y intercepts of the second equation: -4x +y = -11 x-intercept, let y=0 -4x +y = -11 -4x +(0) = -11 -4x = -11 -4 x = -11/-4 = 11/4 x-int = (11/4,0) y-intercept, let x=0 -4x +y = -11 -4(0) + y = -11 y = -11 y-int = (0,-11)

30. Graphing – Method 3 Graph by plotting the x and y intercepts of each line and connecting them to form the line The solution is the intersection: the point (3,1)