SubstitutionSubstitution

Math 374Math 374

TopicsTopics

1) Straight substitution1) Straight substitution2) Point substitution2) Point substitution3) Missing value substitution 3) Missing value substitution 4) Simultaneous substitution 4) Simultaneous substitution

Straight Substitution Straight Substitution

The act of substitution means to The act of substitution means to replace something with something replace something with something elseelse

In mathematics, we will substitute a In mathematics, we will substitute a value for a variable hence changing value for a variable hence changing from an algebraic expression to an from an algebraic expression to an order of operation situation.order of operation situation.

Rules for Order of OperationRules for Order of Operation

1) Inside brackets1) Inside brackets2) Multiplication OR division as they 2) Multiplication OR division as they

occur from left to rightoccur from left to right3) Addition OR subtraction as they 3) Addition OR subtraction as they

occur from left to rightoccur from left to right

SubstitutionSubstitution

Replace the variable with the value Replace the variable with the value in bracketsin brackets

Follow order of operationFollow order of operationEx #1 5x + 2 , x = 7Ex #1 5x + 2 , x = 7= 5 (7) + 2 = 5 (7) + 2 = 35 + 2= 35 + 2= 37= 37

ExercisesExercises

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

Point SubstitutionPoint Substitution

In mathematics, and in this course In mathematics, and in this course we will use a thing call a pointwe will use a thing call a point

For example (6, -9) is a pointFor example (6, -9) is a pointNote open bracket, 1Note open bracket, 1stst number, number,

comma, 2comma, 2ndnd number & close bracket number & close bracketA 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 If we use (6,-9) we mean x=6 and

y=-9y=-9

ExercisesExercises

Consider P = 5x + 2y (9,3)Consider P = 5x + 2y (9,3) This means to substitute x = 9 and y = 3This 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

ExercisesExercises

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

Quiz QuestionsQuiz Questions

1) 3x – 7, x = 21) 3x – 7, x = 22) 5x – 9, x = 42) 5x – 9, x = 43) 25 – 2x, x = 9 3) 25 – 2x, x = 9 4) 2x – 5, x = -34) 2x – 5, x = -35) 9x + 3, x = -115) 9x + 3, x = -11

Quiz QuestionsQuiz Questions

6) 4x + 8, x = -76) 4x + 8, x = -77) 4x + 17, x = -237) 4x + 17, x = -238) 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)

Quiz SolutionsQuiz Solutions

1) -11) -1 2) 112) 11 3) 73) 7 4) -114) -11 5) -965) -96

6) -206) -20 7) -3057) -305 8) -18) -1 9) 439) 43 10) -4910) -49

Missing Value Substitution 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 The (x,5) means we know y = 5, but we do

not know x.not know x. Let us substitute to create an equationLet us substitute to create an equation 5x – 3y =155x – 3y =15 5x – 3 (5) = 155x – 3 (5) = 15 5x – 15 = 155x – 15 = 15 5x = 15 + 155x = 15 + 15 5x = 305x = 30 x = 6x = 6 (6,5)(6,5)

ExercisesExercises

7x – 5y – 35 = 0 (0, y) 7x – 5y – 35 = 0 (0, y) 7 (0) – 5y = 357 (0) – 5y = 35- 5y = 35- 5y = 35y = -7y = -7(0, - 7)(0, - 7)

ExercisesExercises

9y – 3x = 54 (-9 , y)9y – 3x = 54 (-9 , y)9y – 3(-9) = 549y – 3(-9) = 549y + 27 = 549y + 27 = 549y = 279y = 27y = 3y = 3Work in class / Homework do #3a-jWork in class / Homework do #3a-j

Simultaneous SubstitutionSimultaneous Substitution

y = 3x – 5y = 3x – 5y = 5x – 9y = 5x – 9This is a system of equation. Which This is a system of equation. Which

point solves both?point solves both?Based on a mathematical property Based on a mathematical property

called transitive if A = B and A = C called transitive if A = B and A = C what can I say about B & C?what can I say about B & C?

B = CB = C

Simultaneous SubstitutionSimultaneous Substitution

y = 3x – 5 y = 3x – 5 y = 5x – 9y = 5x – 9 3x – 5 = 5x – 93x – 5 = 5x – 9 -2x = -4-2x = -4 x = 2x = 2 We know x but we do not know y. We need to We know x but we do not know y. We need to

substitute back into substitute back into eithereither standard from equation. standard from equation. If x = 2 y = 3x – 5If x = 2 y = 3x – 5 y = 3 (2) -5y = 3 (2) -5 y = 1y = 1 (2,1)(2,1)

ExercisesExercises

y = 4x – 7y = 4x – 7 y = -5x + 20y = -5x + 20 4x – 7 = -5x + 204x – 7 = -5x + 20 9x = 279x = 27 x = 3x = 3 Substitute x = 3 into y = 4x – 7Substitute x = 3 into y = 4x – 7 y = 4 (3) – 7y = 4 (3) – 7 y = 5y = 5 (3, 5) (3, 5)

Exercises Exercises

y = 9x + 22y = 9x + 22 y = 2x + 1y = 2x + 1 9x + 22 = 2x + 19x + 22 = 2x + 1 7x = -217x = -21 x = -3x = -3 x = -3 y = 2x + 1x = -3 y = 2x + 1 y = 2 (-3) + 1y = 2 (-3) + 1 y = -5y = -5 (-3, -5) (-3, -5)

Harder ExercisesHarder Exercises

5x – y = -105x – y = -10 3x – y = -83x – y = -8 Gets harder since you don’t have y= on Gets harder since you don’t have y= on

both lines… get them both into standard both lines… get them both into standard formform

-y = -5x – 10-y = -5x – 10 y = 5x + 10 (that is the 1y = 5x + 10 (that is the 1stst one) one) -y = -3x – 8-y = -3x – 8 y = 3x + 8 (that is the second one) y = 3x + 8 (that is the second one)

Harder Exercises Con’tHarder Exercises Con’t

5x + 10 = 3x + 8 5x + 10 = 3x + 8 2x = -22x = -2x = -1x = -1x = -1 x = -1 y = 5x + 10 y = 5x + 10y = 5 (-1) + 10y = 5 (-1) + 10y = 5y = 5(-1,5)(-1,5)

Harder ExercisesHarder Exercises

4x + y = 14x + y = 1 5x – y = 175x – y = 17 y = -4x + 1y = -4x + 1 -y = -5x + 17-y = -5x + 17 y = 5x – 17y = 5x – 17 -4x + 1 = 5x – 17-4x + 1 = 5x – 17 -9x = -18-9x = -18 x = 2x = 2

x = 2 x = 2 y = -4x + y = -4x + 11

y = -4 (2) + 1y = -4 (2) + 1 y = - 7y = - 7 (2, -7)(2, -7)

Fraction Question Fraction Question

3x + 5y = -143x + 5y = -14 4x – 7y = -54x – 7y = -5 5y = -3x – 145y = -3x – 14 y = y = -3x-3x – – 1414

5 55 5 Don’t you love Don’t you love

fractions?fractions?

-7y = -4x – 5-7y = -4x – 5 -y = -y = -4x – 5-4x – 5

-7-7 y= y= 4x 54x 5

77

Fraction SolutionFraction Solution

-3x – 14-3x – 14 = = 4x + 54x + 5 5 75 7-21x – 98 = 20x + 25-21x – 98 = 20x + 25-41x = 123-41x = 123x = -3x = -33x + 5y = -143x + 5y = -143(-3) + 5y = -143(-3) + 5y = -14-9 + 5y = -14-9 + 5y = -14

Fraction SolutionFraction Solution

5y = -55y = -5y = -1y = -1(-3, -1)(-3, -1)Last one Last one 5x – 7y = 27 5x – 7y = 273x – 2y = 143x – 2y = 14(4,-1)(4,-1)Work in Class / Homework #4 a - oWork in Class / Homework #4 a - o

Work in class / Homework do #4 a-oWork in class / Homework do #4 a-o