Question

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1. john has $0.60 worth of nickels and dimes. he has 4 times as many nickels as dimes. by following the steps below, determine the number of nickels, $x$, and the number of dimes, $y$, that john has.

determine three ways to have 4 times as many nickels as dimes:
(nickels, dimes) = ( __ , __ )
(nickels, dimes) = ( __ , __ )
(nickels, dimes) = ( __ , __ )
on the graph below, plot the three points as if they were coordinate pairs. then draw a line through the points.
determine three ways to have a total value of $0.60:
(nickels, dimes) = ( __ , __ )
(nickels, dimes) = ( __ , __ )
(nickels, dimes) = ( __ , __ )
on the graph below, plot the three points as if they were coordinate pairs. then use them to draw a second line.
use the graph to answer the question:
john has __ nickels and __ dimes.

Explanation:

Step1: Find nickel-dime pairs (4x dimes)

Let dimes = $y$, nickels = $4y$.

• If $y=1$, $x=4$: $(4, 1)$
• If $y=2$, $x=8$: $(8, 2)$
• If $y=3$, $x=12$: $(12, 3)$

Step2: Find pairs worth $0.60

Nickel = $0.05$, dime = $0.10$. Use $0.05x + 0.10y = 0.60$.

• If $y=0$, $x=\frac{0.60}{0.05}=12$: $(12, 0)$
• If $y=3$, $0.05x=0.60-0.30=0.30$, $x=6$: $(6, 3)$
• If $y=6$, $0.05x=0.60-0.60=0$, $x=0$: $(0, 6)$

Step3: Find intersection of two lines

The first line is $x=4y$, the second is $0.05x+0.10y=0.60$. Substitute $x=4y$ into the second equation:
$$0.05(4y) + 0.10y = 0.60$$
$$0.20y + 0.10y = 0.60$$
$$0.30y = 0.60$$
$$y=2$$
Then $x=4(2)=8$.

Answer:

Three (nickels, dimes) pairs with 4x nickels as dimes:

$(4, 1)$, $(8, 2)$, $(12, 3)$

Three (nickels, dimes) pairs worth $0.60:

$(12, 0)$, $(6, 3)$, $(0, 6)$

Final count:

John has 8 nickels and 2 dimes.