2. SIMULTANEOUS EQUATIONS

3. Problem of the Day! n There are 100 animals in a zoo, some which have 2 legs and some have 4 legs. If there are 262 animal legs altogether, how many 4 legged animals are there? Answer: 31 four legged and 69 2 legged!

4. Please note: n This Powerpoint has been uploaded to the wiki: n http://mazenod-10max.wikispaces.com

5. Solving Simultaneous Equations Q. What does the word “simultaneous” mean? A. “at the same time” To solve simultaneous equations you need to have the same number of equations as you have variables (unknowns). Here we will consider cases where we have two equations and two unknowns eg x + y = 3 and 2x – y = 5

6. Substitution Method This method is generally used in situations where one of the variables (say y) is the subject or can easily be made the subject of one (or both) of the equations. Eg 1: y = 3x + 1 and y = 5 – 4x (y is the subject in both) Eg 2: y = 2x - 1 and 2x + 3y = 5 (y is the subject of equation 1) Eg 3: y + 2x = 1 and 2x + 3y = 5 (y can easily be made the subject in equation 1) Eg 4: 3y = 2x - 1 and x + 3y = 5 (x can easily be made the subject of equation 2)

7. Substitution Method In this method we will make use of the mathematical truth: “If things equal the same thing then they must equal each other” ie If a = b and a = c then b = c Example: if y = x+1 and y = p then x+1 = p ie we can swap (substitute) x+1 for y in equation 2 or…

8. Substitution Method Example Label the equations (1) and (2) (1) (2) Step 1 Step 2Make one of the variables (we will use y) the subject of one of the equations (we will use equation 1) From (1) (1’ )

9. Step 3 Step 4 Substitute 3 – x for y in equation (2) (2) Solve the equation Step 5 Substitute this value back into (1’ ) (1’ ) Therefore the solution is (1, 2)

10. What does the solution (1,2) mean? n It means that the point (1,2) is the only solution that satisfies both equations. n x + y = 3  1 + 2 = 3 n x + 5y = 11  1 + 5(2) = 11 n Can you interpret this graphically?

11. Another example Label the equations (1) and (2) (1) (2) Step 1 Step 2 Equation (2) already has x the subject

12. Step 3 Step 4 Substitute 2y + 1 for x in equation 1 (2) Solve the equation Step 5 Substitute this value back into (2 ) Therefore the solution is (3, 1) (1)

13. A more difficult example Solve the simultaneous equations: To get started: Which variable in which equation is the easiest to make the subject? I will choose to make x the subject in equation 2

14. Substitute this for x in equation 1 Therefore the solution is (–4, –2) Substitute y = -2 into (2’) Making x the subject in equation 2 gives: Solve for y

15. Elimination Method This method is best used when you have equations in the form: Label the equations (1) and (2) (1) (2) Step 1

16. Elimination Method Best used when you have equations in the form: Label the equations (1) and (2) (1) (2) Step 1

17. Which variable has the same coefficient? x As the sign of x is the same, we subtract equation (2) from equation (1) Step 2 (1) (2) Step 3 Step 4 x is eliminated, solve for y Step 5 Substitute the value for y into equation (1) and solve for x Therefore the solution is (1, 2)

18. What does the solution mean ? We had two equations and two unknowns, x and y and The solution (1, 2) gives the values of x and y that will make both sentences true.

19. Another example Label the equations (1) and (2) (1) (2) Step 1 Which variable has the same coefficient? y Step 2

20. As the sign of the 3 y is different, we add equation (2) to equation (1) (1) (2) Step 3 Step 4 y is eliminated, solve for x Step 5 Substitute the value for x into equation (1) and solve for y Therefore the solution is (-2, 4)

21. A more difficult example Label the equations (1) and (2) (1) (2) Step 1 Which variable has the same coefficient? Neither Step 2 We can make x have the same coefficient if we multiply equation (2) by 2

22. As the sign is the same, we subtract equation (2’) from equation (1) (1) (2’) Step 3 Step 4 x is eliminated, solve for y Step 5 Substitute the value for y into equation (2) and solve for x Therefore the solution is (-4, 6)

23. Try questions 1 to 4 on the worksheet

24. A more difficult example Using elimination Multiply (1) by 3 and (2) by 4 (1’) (1) (2) (2’) Subtract (2’) from (1’) Substitute into (1) Therefore the solution is (-4, -2)

25. Using substitution From (2) (1) (2) Substitute (2’) into (1) Therefore the solution is (–4, –2) Substitute y = -2 into (2’) (2’)

26. A more difficult example Using elimination Multiply (1) by 3 and (2) by 4 (1’) (1) (2) (2’) Subtract (2’) from (1’) Substitute into (1) Therefore the solution is (-4, -2)

27. Using substitution From (2) (1) (2) Substitute (2’) into (1) Therefore the solution is (–4, –2) Substitute y = -2 into (2’) (2’)

28. Step 3 Step 4 Substitute the new equation into the other equation in this case equation (2) (2) Solve the equation Step 5 Substitute this value back into (1’ ) (1’ ) Therefore the solution is (1, 2)

29. Step 3 Step 4 Substitute equation (2) into equation (1) (2) Solve the equation Step 5 Substitute this value back into (2 ) Therefore the solution is (3, 1) (1)

30. Complete the questions on the worksheet

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