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## Discussion

## Study Guides

## Practice Questions

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## Table of Contents

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### Solving by Substituting

- To find exact solutions, use algebraic methods like the
*substitution method*. - Use substitution when at least one of the coefficients in the system is 1 or –1. Solve for this variable in terms of the other one. Then substitute that expression into the other equation.
- If you eventually get an equation that is always true, then the system has an infinite number of solutions.
- If you eventually get an equation that is never true, then the system has no solution.

### Solving by Substituting

j = 6 − 11k3j − 7k = 92

- 3(6 − 11k) − 7k = 92
- 18 − 33k − 7k = 92
- − 40k = 74
- k = − [74/( − 40)]

k = − [37/20]

5m + 6n = − 299m − n = 6

- − n = 6 − 9m
- n = − 6 + 9m
- 5m + 6( − 6 − 9m) = − 29
- 5m − 36 − 54m = − 29
- − 49m = 7
- m = - [1/7]
- 9 ( − [1/7] ) − n = 6
- − [9/7] − n = 6
- − n = 7[2/7]

n = − 7[2/7]

m = − [1/7]

m = − [1/7]

8a + 2b = 104a − 6b = − 23

- Let's start by solving for b in the first equation: 8a + 2b = 10
- 2b = 10 − 8a
- b = 5 − 4a
- Plug b into the other equation 4a − 6(5 − 4a) = − 23
- 4a − 30 + 24a = − 23
- 28a = 7
- a = [7/28] = [1/4]
- 4g( [1/4] ) − 6b = − 23
- 1 − 6b = − 23
- − 6b = − 24
- b = 4

a = [1/4]

b=4

b=4

x = 4y + 95x + 3y = 14

- 5(4y + 9) + 3y = 14
- 20y + 45 + 3y = 14
- 23y + 45 = 14
- 23y = − 31
- y = − [31/23]
- x = 4y + 9x = 4( − [31/23] ) + 9
- x = − [124/23] + 9
- x = − 115[9/23] + 9

x = - 106[9/23]

x = 5y + 217x − 3y = 16

- 7(5y + 21) − 3y = 16
- 35y + 147 − 3y = 16
- 32y + 147 = 16
- 32y = − 131
- y = − [131/32]
- x = 5( − [131/32] ) + 21
- x = − 20[15/32] + 21

x = [17/32]

x = 4y − 75x − 3y = 15

- 5(4y − 7) − 3y = 15
- 20y − 35 − 3y = 15
- 17y − 35 = 15
- 17y = 50
- y = [50/17] = 2[16/17]
- x = 4( [50/17] ) − 7
- x = [200/17] − 7

x = 4[13/17]

x − 3y = 62x − 5y = 10

- x = 6 + 3y
- 2(6 + 3y) − 5y = 10
- 12 + 6y − 5y = 10
- 12 + y = 10
- y = − 2
- x = 6 + 3yx = 6 + 3( − 2)
- x = 6 − 6

x = 0

5x − 6y = 12x − 2y = 8

- x = 8 + 2y
- 5(8 + 2y) − 6y = 12
- 40 + 10y − 6y = 12
- 40 + 4y = 12
- 4y = − 28
- y = − 7
- x = 8 + 2yx = 8 + 2( − 7)
- x = 8 − 14

x = − 6

24x − 4y = 603x − y = 12

- − y = − 3x + 12
- y = 3x − 12
- 24x − 4(3x − 12) = 60
- 24x − 12x + 48 = 60
- 12x + 48 = 60
- 12x = 12
- x = 2
- y = 3x − 12y = 3(2) − 12
- y = 6 − 12

y = − 6

8x − 6y = 143x + 3y = 9

- 3x + 3y = 9
- 3x = 9 − 3y
- x = [(9 − 3y)/3]
- x = 3 − y
- 8x − 6y = 148(3 − y) − 6y = 14
- 24 − 8y − 6y = 14
- 24 − 14y = 14
- − 14y = − 10
- y = [( − 10)/( − 14)] = [5/7]
- x = 3 − yx = 3 − [5/7]

x = 2[2/7]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Solving by Substituting

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Substitution 0:09
- Example
- Number of Solutions 2:47
- Infinite Solutions
- No Solutions
- Example 1: Solve by Substitution 5:44
- Example 2: Solve by Substitution 10:01
- Example 3: Solve by Substitution 15:17
- Example 4: Solve by Substitution 19:41

0 answers

Post by Michael Romanov on October 15, 2016

Example II from the lecture:

y equals -13/7 not 13/7, 36 - 49 is -13

1 answer

Last reply by: Dr Carleen Eaton

Sat Apr 27, 2013 1:56 PM

Post by Shoshana Coleman on April 22, 2013

I'm a little confused, in example 3 you said not to substitute the x back into itself but to substitute it into the other equation , does that apply for every substitution method (that you can not plug in the substituted x into the same equation)?

0 answers

Post by bo young lee on April 10, 2013

where can i find more information about substitution???

2 answers

Last reply by: Velmurugan Gurusamy

Fri Dec 14, 2012 7:31 AM

Post by Nagayasu Toshitatsu on September 24, 2012

In example II, I believe that you forgot the negative in y. y should be -7/13.

1 answer

Last reply by: Dr Carleen Eaton

Mon Feb 7, 2011 5:31 PM

Post by shah Mahmoodi on April 6, 2010

X= 31\44 is wrong because the dominator should be 31\11