Respuesta :
Part a)
3x+2y=12------> equation 1
y=x-9------> equation 2
substitute equation 2 in equation 1
3x+2*[x-9]=12-----> 3x+2x-18=12----> 5x=30------> x=6
y=6-9-----> y=-3
the answer part a) is
(6,-3)-------> word-----> HE
Part b)
4x+y=-2-----> equation 1
y=2x-2-----> equation 2
substitute equation 2 in equation 1
4x+[2x-2]=-2-----> 6x=0----> x=0
y=2*0-2-----> y=-2
the answer part b) is
(0,-2)-------> word-----> MADE
Part c)
-3x+5y=5-----> equation 1
y=x-1------> equation 2
substitute equation 2 in equation 1
-3x+5*[x-1]=5-----> -3x+5x-5=5----> 2x=10-----> x=5
y=5-1-----> y=4
the answer Part c) is
(5,4)-------> word-----> A
Part d)
2x+y=-16-----> equation 1
y=2x---> equation 2
substitute equation 2 in equation 1
2x+[2x]=-16-----> 4x=-16-----> x=-4
y=2*(-4)-----> y=-8
the answer Part d) is
(-4,-8)-------> word-----> KNOT
Part e)
7x=-35------> x=-5
-8x+9y=4-----> -8*(-5)+9y=4-----> 40+9y=4----> 9y=-36-----> y=-4
the answer Part e) is
(-5,-4)-------> word-----> IN
Part f)
-4x+3y=20-----> equation 1
-14y=-56----> y=4
substitute in equation 1
-4x+3*4=20-----> -4x=8----> x=-2
the answer Part f) is
(-2,4)-------> word-----> HIS
Part g)
13x-6y=-5------> equation 1
x+10=11-----> x=1
13*1-6y=-5------> -6y=-5-13-----> y=3
the answer Part g) is
(1,3)-------> word-----> TAIL
Part h)
9x-2y=12----> equation 1
y+4=16------> y=12
9x-2*12=12-----> 9x=36----> x=4
the answer Part h) is
(4,12)-------> word-----> AND
Part i)
x=6+2y------> equation 1
-3x+14y=-18------> equation 2
substitute equation 1 in equation 2
-3*[6+2y]+14y=-18------> -18-6y+14y=-18----> 8y=0-----> y=0
x=6+2*0-----> x=6
the answer Part i) is
(6,0)-------> word-----> CALLED
Part j)
5x-9y=12-----> equation 1
y=-6-x-----> equation 2
substitute equation 2 in equation 1
5x-9*[-6-x]=12-------> 5x+54+9x=12-----> 14x=-42------> x=-3
y=6-(-3)----> y=-3
the answer Part j) is
(-3,-3)-------> word-----> IT
Part k)
x=-8+y------> equation 1
6x+y=-6------> equation 2
substitute equation 1 in equation 2
6*[-8+y]+y=-6-----> -48+6y+y=-6-----> 7y=42-----> y=6
x=-8+6-----> x=-2
the answer Part k) is
(-2,6)-------> word-----> A
Part l)
7x-3y=17----> equation 1
y=2x-6------> equation 2
substitute equation 2 in equation 1
7x-3*[2x-6]=17------> 7x-6x+18=17-----> x=-1
y=2*(-1)-6----> y=-8
the answer Part l) is
(-1,-8)-------> word-----> PIGS
part m)
The physical education instructor asked each student to do a total of 36 pull-ups and push-ups in 1 minute. The instructor wanted students to do 8 times as many push-ups as pull-ups. Write a system of linear equations that represents this situation. How many pull-ups and push-ups were required in 1 minute?
let
x-------->pull-ups
y------->push-ups
we know thatx+y=36-----> equation 1
y=8x-----> equation 2
substitute equation 2 in equation 1
x+8x=36--------> 9x=36------> x=4
y=8*4-----> y=32
the solution is (4.32)---------> word TIE
HE MADE A KNOT IN HIS TAIL AND CALLED IT A PIG'S TIE
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
[tex]3x + 2(x - 9) = 12\\3x + 2x - 18 = 12\\5x - 18 = 12\\5x = 12 + 18\\5x = 30\\x = 6[/tex]
- Substitute x = 6 into eqn. 2 to find y
[tex]y = 6 - 9\\y = -3[/tex]
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
[tex]4x + 2x - 2 = -2\\6x - 2 = -2\\6x = -2 + 2\\6x = 0\\x = 0[/tex]
- Substitute x = 0 into eqn. 2 to find y
[tex]y = 2(0) - 2 \\y = 0 - 2\\y = -2[/tex]
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
[tex]-3x + 5(x - 1) = 5\\-3x + 5x - 5 = 5\\2x - 5 = 5\\2x = 5 + 5\\2x = 10\\x = 5[/tex]
- Substitute x = 5 into eqn. 2 to find y
[tex]y = 5 - 1\\y = 4[/tex]
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
[tex]2x + 2x = -16\\4x = -16\\x = -4[/tex]
- Substitute x = -4 into eqn. 2 to find y
[tex]y = 2(-4)\\y = -8[/tex]
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
[tex]7x = -35\\x = -5[/tex]
- Substitute x = -5 into eqn. 2 to find y
[tex]-8(-5) + 9y = 4\\40 + 9y = 4\\9y = 4 - 40\\9y = -36\\y = -4[/tex]
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
[tex]-14y = -56\\y = 4[/tex]
- Substitute y = 4 into eqn. 1 to find y
[tex]-4x + 3(4) = 20\\-4x + 12 = 20\\-4x = 20 - 12\\-4x = 8\\x = -2[/tex]
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
[tex]13(1) - 6y = -5 \\13 - 6y = -5\\-6y = -5 -13\\-6y = -18\\y = 3[/tex]
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
[tex]-3(6 + 2y) + 14y = -18\\-18 - 6y + 14y = -18\\8y = -18 + 18\\8y = 0\\y = 0[/tex]
- Substitute y = 0 into eqn. 1 to find x
[tex]x = 6 + 2(0)\\x = 6[/tex]
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
[tex]9x - 2(12) = 12\\9x - 24 = 12\\9x = 12 + 24\\9x = 36\\x = 4[/tex]
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
[tex]-3(6 + 2y) + 14y = -18\\-18 - 6y + 14y = -18\\8y = -18 + 18\\8y = 0\\y = 0[/tex]
- Substitute y = 0 into eqn. 1 to find x
[tex]x = 6 + 2(0)\\x = 6[/tex]
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
[tex]5x - 9(-6 - x) = 12\\5x + 54 + 9x = 12\\14x = 12 - 54\\14x = -42\\x = -3[/tex]
- Substitute x = -3 into eqn. 2 to find y
[tex]y = -6 - (-3)\\y = -3[/tex]
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
[tex]6(-8 + y) + y = -6\\-48 + 6y + y = -6\\7y = -6 + 48\\7y = 42\\y = 6[/tex]
- 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
[tex]7x - 3(2x - 6) = 17\\7x - 6x + 18 = 17\\x = 17 - 18\\x = -1[/tex]
- Substitute x = -1 into eqn. 2 to find y
[tex]y = 2(-1) - 6\\y = -2 - 6\\y = -8[/tex]
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-ups
- y = 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
[tex]x + 8x = 36\\\\9x = 36\\\\x = 4[/tex]
Number of push-ups in 1 min = 4
- Substitute x = 4 into eqn. 2 to find the value of y
[tex]y = 8(4)\\\\y = 32[/tex]
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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