By solving the systems of linear equations given by substitution, the riddle solved is: HE MADE A KNOT IN HIS TAIL AND CALLED IT A PIG'S TIE
To solve the riddle, the following systems of linear equations would be solved by finding the values of x and y by substitution:
A. 3x + 2y = 12 --> Eqn. 1
y = x - 9 --> Eqn. 2
Substitute y = x - 9 into eqn. 1 to find x
![3x + 2(x - 9) = 12\\3x + 2x - 18 = 12\\5x - 18 = 12\\5x = 12 + 18\\5x = 30\\x = 6](/tpl/images/0092/8052/05a3c.png)
Substitute x = 6 into eqn. 2 to find y
![y = 6 - 9\\y = -3](/tpl/images/0092/8052/5384a.png)
The solution is (6, -3) which tallies with the word: HE
B. 4x + y = -2 --> Eqn. 1
y = 2x - 2 --> Eqn. 2
Substitute y = 2x - 2 into eqn. 1 to find x
![4x + 2x - 2 = -2\\6x - 2 = -2\\6x = -2 + 2\\6x = 0\\x = 0](/tpl/images/0092/8052/64f48.png)
Substitute x = 0 into eqn. 2 to find y
![y = 2(0) - 2 \\y = 0 - 2\\y = -2](/tpl/images/0092/8052/2e322.png)
The solution is (0, -2) which tallies with the word: MADE
C. -3x + 5y = 5 --> Eqn. 1
y = x - 1 --> Eqn. 2
Substitute y = x - 1 into eqn. 1 to find x
![-3x + 5(x - 1) = 5\\-3x + 5x - 5 = 5\\2x - 5 = 5\\2x = 5 + 5\\2x = 10\\x = 5](/tpl/images/0092/8052/f221d.png)
Substitute x = 5 into eqn. 2 to find y
![y = 5 - 1\\y = 4](/tpl/images/0092/8052/08bef.png)
The solution is (5, 4) which tallies with the word: A
D. 2x + y = -16 --> Eqn. 1
y = 2x --> Eqn. 2
Substitute y = 2x into eqn. 1 to find x
![2x + 2x = -16\\4x = -16\\x = -4](/tpl/images/0092/8052/92502.png)
Substitute x = -4 into eqn. 2 to find y
![y = 2(-4)\\y = -8](/tpl/images/0092/8052/b5438.png)
The solution is (-4, -8) which tallies with the word: KNOT
E. 7x = -35 --> Eqn. 1
-8x + 9y = 4 --> Eqn. 2
Solve eqn. 1. to find x
![7x = -35\\x = -5](/tpl/images/0092/8052/f7313.png)
Substitute x = -5 into eqn. 2 to find y
![-8(-5) + 9y = 4\\40 + 9y = 4\\9y = 4 - 40\\9y = -36\\y = -4](/tpl/images/0092/8052/9c8b5.png)
The solution is (-5, -4) which tallies with the word: IN
F. -4x + 3y = 20 --> Eqn. 1
-14y = -56 --> Eqn. 2
Solve eqn. 2 to find y
![-14y = -56\\y = 4](/tpl/images/0092/8052/7e972.png)
Substitute y = 4 into eqn. 1 to find y
![-4x + 3(4) = 20\\-4x + 12 = 20\\-4x = 20 - 12\\-4x = 8\\x = -2](/tpl/images/0092/8052/adecb.png)
The solution is (-2, 4) which tallies with the word: HIS
G. 13x - 6y = -5 --> Eqn. 1
x + 10 = 11 --> Eqn. 2
Solve eqn. 2 to find x
x = 11 - 10
x = 1
Substitute x = 1 into eqn. 1 to find y
![13(1) - 6y = -5 \\13 - 6y = -5\\-6y = -5 -13\\-6y = -18\\y = 3](/tpl/images/0092/8052/9d49d.png)
The solution is (1, 3) which tallies with the word: TAIL
i. x = 6 + 2y --> Eqn. 1
-3x + 14y = -18 --> Eqn. 2
Substitute x = 6 + 2y into eqn. 2 to find y
![-3(6 + 2y) + 14y = -18\\-18 - 6y + 14y = -18\\8y = -18 + 18\\8y = 0\\y = 0](/tpl/images/0092/8052/54dd9.png)
Substitute y = 0 into eqn. 1 to find x
![x = 6 + 2(0)\\x = 6](/tpl/images/0092/8052/bb6a4.png)
The solution is (6, 0) which tallies with the word: CALLED
H. 9x - 2y = 12 --> Eqn. 1
y + 4 = 16 --> Eqn. 2
Solve eqn. 2 to find y
y = 16 - 4
y = 12
Substitute y = 12 into eqn. 1 to find x
![9x - 2(12) = 12\\9x - 24 = 12\\9x = 12 + 24\\9x = 36\\x = 4](/tpl/images/0092/8052/0a432.png)
The solution is (4, 12) which tallies with the word: AND
I. x = 6 + 2y --> Eqn. 1
-3x + 14y = -18 --> Eqn. 2
Substitute x = 6 + 2y into eqn. 2 to find y
![-3(6 + 2y) + 14y = -18\\-18 - 6y + 14y = -18\\8y = -18 + 18\\8y = 0\\y = 0](/tpl/images/0092/8052/54dd9.png)
Substitute y = 0 into eqn. 1 to find x
![x = 6 + 2(0)\\x = 6](/tpl/images/0092/8052/bb6a4.png)
The solution is (6, 0) which tallies with the word: CALLED
J. 5x - 9y = 12 --> Eqn. 1
y = -6 - x --> Eqn. 2
Substitute y = -6 - x into eqn. 1 to find x
![5x - 9(-6 - x) = 12\\5x + 54 + 9x = 12\\14x = 12 - 54\\14x = -42\\x = -3](/tpl/images/0092/8052/42350.png)
Substitute x = -3 into eqn. 2 to find y
![y = -6 - (-3)\\y = -3](/tpl/images/0092/8052/d4e14.png)
The solution is (-3, -3) which tallies with the word: IT
K. x = -8 + y--> Eqn. 1
6x + y = -6 --> Eqn. 2
Substitute x = -8 + y into eqn. 2 to find y
![6(-8 + y) + y = -6\\-48 + 6y + y = -6\\7y = -6 + 48\\7y = 42\\y = 6](/tpl/images/0092/8052/87412.png)
Substitute y = 6 into eqn. 1 to find x
x = -8 + 6
x = -2
The solution is (-2, 6) which tallies with the word: A
L. 7x - 3y = 17 --> Eqn. 1
y = 2x - 6 --> Eqn. 2
Substitute y = 2x - 6 into eqn. 1 to find x
![7x - 3(2x - 6) = 17\\7x - 6x + 18 = 17\\x = 17 - 18\\x = -1](/tpl/images/0092/8052/e1881.png)
Substitute x = -1 into eqn. 2 to find y
![y = 2(-1) - 6\\y = -2 - 6\\y = -8](/tpl/images/0092/8052/0b91b.png)
The solution is (-1, -8) which tallies with the word: PIGS
M. Given that the total number of push-ups and pull-ups equals 36,
Let,
x = number of push-upsy = number of pull-ups
The first equation that can be formed from this is:
x + y = 36 --> Eqn. 1
Also, if the instructor wants 8 times as many push-ups as pull-ups from the students, we would have the second equation as:
y = 8x --> Eqn. 2
Substitute y = 8x into eqn. 1 to find the value of x
![x + 8x = 36\\\\9x = 36\\\\x = 4](/tpl/images/0092/8052/32d5f.png)
Number of push-ups in 1 min = 4
Substitute x = 4 into eqn. 2 to find the value of y
![y = 8(4)\\\\y = 32](/tpl/images/0092/8052/26af8.png)
Number of push-ups in 1 min = 32
The solution to the system of linear equations would be (4, 32) which corresponds to the word: TIE.
Therefore, by solving the systems of linear equations given by substitution, the riddle solved is: HE MADE A KNOT IN HIS TAIL AND CALLED IT A PIG'S TIE
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