Answer: There are 12 pennies, 96 nickels, and 5 quarters.
Step-by-step explanation:
Let x = number of pennies
y= number of nickels
z= number of quarters
As per given ,[tex]x+y+z = 113\ \ ...(i)[/tex]
[tex]y=8x\ \ ...(ii)[/tex]
Since 1 penny =1 cent
1 nickel = 5 cents
1 quarter = 25 cents
so, $6.17 = 617cents [1 dollar = 100 cents]
Now, [tex]x+5y+25z= 617...(iii)[/tex]
Required system of linear equations:
[tex]x+y+z = 113\ \ ...(i)[/tex]
[tex]y=8x\ \ ...(ii)[/tex]
[tex]x+5y+25z= 617...(iii)[/tex]
Put value of y from (ii) in (i), we get
[tex]x+8x+z=113\\\\\Rightarrow\ 9x+z=113\\\\\Rightarrow z=113-9x\ \ (iv)[/tex]
Use (ii) and (iv) in (iii), we get
[tex]x+10(8x)+25(113-9x)=617[/tex]
[tex]\Rightarrow\ x+5\cdot \:8x+25\left(113-9x\right)=617\\\\\Rightarrowx+40x+25\left(-9x+113\right)=617\\\\\Rightarrow-184x+2825=617\\\\\Rightarrow-184x+2825-2825=617-2825\\\\\Rightarrow-184x=-2208\\\\\Rightarrow\frac{-184x}{-184}=\frac{-2208}{-184}\\\\\Rightarrow x=12[/tex]
from (ii), y= 8(12)=96
from (iv), z=113-9(12)=113-108= 5
hence, there are 12 pennies, 96 nickels, and 5 quarters.
Using a system of equations, it is found that there are 12 pennies, 96 nickels and 5 quarters.
In the system of equations, I am going to say that:
There is a total of 113 coins, hence:
[tex]x + y + z = 113[/tex]
$6.17 in change, hence, considering the value of each coin:
[tex]0.01x + 0.05y + 0.25z = 6.17[/tex]
8 times as many nickels as pennies, hence:
[tex]y = 8x[/tex]
Replacing the third equation on the first two:
[tex]x + y + z = 113[/tex]
[tex]x + 8x + z = 113[/tex]
[tex]9x + z = 113[/tex]
[tex]z = 113 - 9x[/tex]
Then
[tex]0.01x + 0.05y + 0.25z = 6.17[/tex]
[tex]0.01x + 0.05(8x) + 0.25(113 - 9x) = 6.17[/tex]
[tex]-1.84x = -22.08[/tex]
[tex]1.84x = 22.08[/tex]
[tex]x = \frac{22.08}{1.84}[/tex]
[tex]x = 12[/tex]
Then
[tex]y = 8x = 8(12) = 96[/tex]
[tex]z = 113 - 9x = 113 - 9(12) = 5[/tex]
There are 12 pennies, 96 nickels and 5 quarters.
To learn more about system of equations, you can take a look at https://brainly.com/question/9553406
