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
 In mathematics, we will substitute a value
for a variable hence changing from an
algebraic expression to an order of
operation situation.
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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
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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
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Exercises
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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
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
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Exercises
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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
Exercises
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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 
Quiz Questions
 1)
 2)
 3)
 4)
 5)
3x – 7, x = 2
5x – 9, x = 4
25 – 2x, x = 9
2x – 5, x = -3
9x + 3, x = -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)
Quiz Solutions
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1)
2)
3)
4)
5)
-1
11
7
-11
-96
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6) -20
7) -305
8) -1
9) 43
10) -49
Missing Value Substitution
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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)
Exercises
7x – 5y – 35 = 0 (0, y)
 7 (0) – 5y = 35
 - 5y = 35
 y = -7
 (0, - 7)
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Exercises
9y – 3x = 54 (-9 , y)
 9y – 3(-9) = 54
 9y + 27 = 54
 9y = 27
y=3
 Work in class / Homework do #3a-j
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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
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Simultaneous Substitution
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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)
Exercises
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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)
Exercises
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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)
Harder Exercises
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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)
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)
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Harder Exercises
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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)
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Fraction Question 
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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
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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
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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
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Work in class / Homework do #4 a-o