Substitution Math 374 Topics l 1 Straight substitution

Substitution Math 374

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Simultaneous Substitution l l l l l y = 3 x – 5 y = 5 x – 9 3 x – 5 = 5 x – 9 -2 x = -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 = 3 x – 5 y = 3 (2) -5 y=1 (2, 1)

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

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

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

Harder Exercises Con’t l 5 x + 10 = 3 x + 8 l 2 x = -2 l x = -1 y = 5 x + 10 l y = 5 (-1) + 10 ly = 5 l (-1, 5)

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

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

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

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

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