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Substitution

Substitution. Math 374. Topics. 1) Straight substitution 2) Point substitution 3) Missing value substitution 4) Simultaneous substitution. Straight Substitution. The act of substitution means to replace something with something else

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Substitution

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  1. Substitution Math 374

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

  3. Straight Substitution • 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.

  4. Rules for Order of Operation • 1) Inside brackets • 2) Multiplication OR division 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 • Follow order of operation • Ex #1 5x + 2 , x = 7 • = 5 (7) + 2 • = 35 + 2 • = 37

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

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

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

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

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

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

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

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

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

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

  16. Simultaneous Substitution • y = 3x – 5 • y = 5x – 9 • 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? • B = C

  17. Simultaneous Substitution • y = 3x – 5 • y = 5x – 9 • 3x – 5 = 5x – 9 • -2x = -4 • x = 2 • 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 • y = 3 (2) -5 • y = 1 • (2,1)

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

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

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

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

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

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

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

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

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

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