1. Substitution Math 374

2. Topics 1) Straight substitution 1) Straight substitution 2) Point substitution 2) Point substitution 3) Missing value substitution 3) Missing value substitution 4) Simultaneous substitution 4) Simultaneous substitution

3. Straight Substitution The act of substitution means to replace something with something else The act of substitution means to replace something with something else In mathematics, we will substitute a value for a variable hence changing from an algebraic expression to an order of operation situation. In mathematics, we will substitute a value for a variable hence changing from an algebraic expression to an order of operation situation.

4. Rules for Order of Operation 1) Inside brackets 1) Inside brackets 2) Multiplication OR division as they occur from left to right 2) Multiplication OR division as they occur from left to right 3) Addition OR subtraction as they occur from left to right 3) Addition OR subtraction as they occur from left to right

5. Substitution Replace the variable with the value in brackets Replace the variable with the value in brackets Follow order of operation Follow order of operation Ex #1 5x + 2, x = 7 Ex #1 5x + 2, x = 7 = 5 (7) + 2 = 5 (7) + 2 = 35 + 2 = 35 + 2 = 37 = 37

6. Exercises Ex #2: 3x – 9, x = 4 Ex #2: 3x – 9, x = 4 = 3 (4) – 9 = 3 (4) – 9 = 12 – 9 = 12 – 9 = 3 = 3 Ex #3: 3x – 11, x = -5 Ex #3: 3x – 11, x = -5 = 3 (-5) – 11 = 3 (-5) – 11 = -15 – 11 = -15 – 11 = - 26 = - 26 Work in class / Homework #1 a - t Work in class / Homework #1 a - t

7. Point Substitution In mathematics, and in this course we will use a thing call a point In mathematics, and in this course we will use a thing call a point For example (6, -9) is a point For example (6, -9) is a point Note open bracket, 1 st number, comma, 2 nd number & close bracket Note open bracket, 1 st number, comma, 2 nd number & close bracket A point always has the format of (x,y) A point always has the format of (x,y) If we use (6,-9) we mean x=6 and y=-9 If we use (6,-9) we mean x=6 and y=-9

8. Exercises Consider P = 5x + 2y (9,3) Consider P = 5x + 2y (9,3) This means to substitute x = 9 and y = 3 This means to substitute x = 9 and y = 3 P = 5 (9) + 2 (3) P = 5 (9) + 2 (3) = 45 + 6 = 45 + 6 = 51 = 51 K = 9x – 3y (-2, 4) K = 9x – 3y (-2, 4) = 9 (-2) – 3 (4) = 9 (-2) – 3 (4) -18 – 12 -18 – 12 -30 -30

9. Exercises Q = 7x – 5y – 3 (-1, - 7) Q = 7x – 5y – 3 (-1, - 7) = 7 (-1) – 5 (-7) - 3 = 7 (-1) – 5 (-7) - 3 = -7 + 35 – 3 = -7 + 35 – 3 = 25 = 25 T = 5y – 3x – 7 (-2, -5) T = 5y – 3x – 7 (-2, -5) = 5 (-5) – 3 (-2) – 7 = 5 (-5) – 3 (-2) – 7 = -25 + 6 – 7 = -25 + 6 – 7 = -26 = -26 Work in class / Homework: Do #2 a – o Work in class / Homework: Do #2 a – o Quiz Quiz

10. Quiz Questions 1) 3x – 7, x = 2 1) 3x – 7, x = 2 2) 5x – 9, x = 4 2) 5x – 9, x = 4 3) 25 – 2x, x = 9 3) 25 – 2x, x = 9 4) 2x – 5, x = -3 4) 2x – 5, x = -3 5) 9x + 3, x = -11 5) 9x + 3, x = -11

11. Quiz Questions 6) 4x + 8, x = -7 6) 4x + 8, x = -7 7) 4x + 17, x = -23 7) 4x + 17, x = -23 8) P = 9x – 7y (3,4) 8) P = 9x – 7y (3,4) 9) 8x – 9y (2,-3) 9) 8x – 9y (2,-3) 10) 5x – 3y (-5,8) 10) 5x – 3y (-5,8)

12. Quiz Solutions 1) -1 1) -1 2) 11 2) 11 3) 7 3) 7 4) -11 4) -11 5) -96 5) -96 6) -20 6) -20 7) -305 7) -305 8) -1 8) -1 9) 43 9) 43 10) -49 10) -49

13. Missing Value Substitution Consider 5x – 3y = 15 (x,5) Consider 5x – 3y = 15 (x,5) The (x,5) means we know y = 5, but we do not know x. The (x,5) means we know y = 5, but we do not know x. Let us substitute to create an equation Let us substitute to create an equation 5x – 3y =15 5x – 3y =15 5x – 3 (5) = 15 5x – 3 (5) = 15 5x – 15 = 15 5x – 15 = 15 5x = 15 + 15 5x = 15 + 15 5x = 30 5x = 30 x = 6 x = 6 (6,5) (6,5)

14. Exercises 7x – 5y – 35 = 0 (0, y) 7x – 5y – 35 = 0 (0, y) 7 (0) – 5y = 35 7 (0) – 5y = 35 - 5y = 35 - 5y = 35 y = -7 y = -7 (0, - 7) (0, - 7)

15. Exercises 9y – 3x = 54 (-9, y) 9y – 3x = 54 (-9, y) 9y – 3(-9) = 54 9y – 3(-9) = 54 9y + 27 = 54 9y + 27 = 54 9y = 27 9y = 27 y = 3 y = 3 Work in class / Homework do #3a-j Work in class / Homework do #3a-j

16. Simultaneous Substitution y = 3x – 5 y = 3x – 5 y = 5x – 9 y = 5x – 9 This is a system of equation. Which point solves both? This is a system of equation. Which point solves both? Based on a mathematical property called transitive if A = B and A = C what can I say about B & C? Based on a mathematical property called transitive if A = B and A = C what can I say about B & C? B = C B = C

17. Simultaneous Substitution y = 3x – 5 y = 3x – 5 y = 5x – 9 y = 5x – 9 3x – 5 = 5x – 9 3x – 5 = 5x – 9 -2x = -4 -2x = -4 x = 2 x = 2 We know x but we do not know y. We need to substitute back into either standard from equation. We know x but we do not know y. We need to substitute back into either standard from equation. If x = 2 y = 3x – 5 If x = 2 y = 3x – 5 y = 3 (2) -5 y = 3 (2) -5 y = 1 y = 1 (2,1) (2,1)

18. Exercises y = 4x – 7 y = 4x – 7 y = -5x + 20 y = -5x + 20 4x – 7 = -5x + 20 4x – 7 = -5x + 20 9x = 27 9x = 27 x = 3 x = 3 Substitute x = 3 into y = 4x – 7 Substitute x = 3 into y = 4x – 7 y = 4 (3) – 7 y = 4 (3) – 7 y = 5 y = 5 (3, 5) (3, 5)

19. Exercises y = 9x + 22 y = 9x + 22 y = 2x + 1 y = 2x + 1 9x + 22 = 2x + 1 9x + 22 = 2x + 1 7x = -21 7x = -21 x = -3 x = -3 x = -3 y = 2x + 1 x = -3 y = 2x + 1 y = 2 (-3) + 1 y = 2 (-3) + 1 y = -5 y = -5 (-3, -5) (-3, -5)

20. Harder Exercises 5x – y = -10 5x – y = -10 3x – y = -8 3x – y = -8 Gets harder since you don’t have y= on both lines… get them both into standard form Gets harder since you don’t have y= on both lines… get them both into standard form -y = -5x – 10 -y = -5x – 10 y = 5x + 10 (that is the 1 st one) y = 5x + 10 (that is the 1 st one) -y = -3x – 8 -y = -3x – 8 y = 3x + 8 (that is the second one) y = 3x + 8 (that is the second one)

21. Harder Exercises Con’t 5x + 10 = 3x + 8 5x + 10 = 3x + 8 2x = -2 2x = -2 x = -1 x = -1 x = -1  y = 5x + 10 x = -1  y = 5x + 10 y = 5 (-1) + 10 y = 5 (-1) + 10 y = 5 y = 5 (-1,5) (-1,5)

22. Harder Exercises 4x + y = 1 4x + y = 1 5x – y = 17 5x – y = 17 y = -4x + 1 y = -4x + 1 -y = -5x + 17 -y = -5x + 17 y = 5x – 17 y = 5x – 17 -4x + 1 = 5x – 17 -4x + 1 = 5x – 17 -9x = -18 -9x = -18 x = 2 x = 2 x = 2  y = -4x + 1 x = 2  y = -4x + 1 y = -4 (2) + 1 y = -4 (2) + 1 y = - 7 y = - 7 (2, -7) (2, -7)

23. Fraction Question Fraction Question 3x + 5y = -14 3x + 5y = -14 4x – 7y = -5 4x – 7y = -5 5y = -3x – 14 5y = -3x – 14 y = -3x – 14 y = -3x – 14 5 5 5 5 Don’t you love fractions? Don’t you love fractions? -7y = -4x – 5 -7y = -4x – 5 -y = -4x – 5 -y = -4x – 5 -7 -7 y= 4x 5 y= 4x 5 7

24. Fraction Solution -3x – 14 = 4x + 5 -3x – 14 = 4x + 5 5 7 5 7 -21x – 98 = 20x + 25 -21x – 98 = 20x + 25 -41x = 123 -41x = 123 x = -3 x = -3 3x + 5y = -14 3x + 5y = -14 3(-3) + 5y = -14 3(-3) + 5y = -14 -9 + 5y = -14 -9 + 5y = -14

25. Fraction Solution 5y = -5 5y = -5 y = -1 y = -1 (-3, -1) (-3, -1) Last one 5x – 7y = 27 Last one 5x – 7y = 27 3x – 2y = 14 3x – 2y = 14 (4,-1) (4,-1) Work in Class / Homework #4 a - o Work in Class / Homework #4 a - o

26. Work in class / Homework do #4 a-o Work in class / Homework do #4 a-o