**Q = 53**[divide both sides by (-20)]

**N = 27**[divide both sides by (-1)]

**Q = 53**

**N = 27**

Using elimination or subsistution

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Choose appropriate variables (either x and y, or N and Q, or other). Let:

N = number of nickels

Q = number of quarters

Now, realize that the value of the nickels is 5N and the value of the quarters is 25Q.

Translate:

"containing 80 coins " means N+Q = 80 [eq1]

"total value of the coins is $14.60" means 5N + 25Q = 1460 [eq2]

[note: It is easier to use cents than to keep the decimal point]

The math:

(1) elimination - the coefficient of one of the variables must be the same (or opposite) to subtract (or add) the equations.

5N + 5Q = 400 [5*eq1]

5N + 25Q = 1460 [eq2]

---------------------------------- [elimination; subtract equations]

-20Q = -1060

N + Q = 80 [eq1 again]

------------------------------- [elimination; subtract equations]

-N = -27

Check:

Is 5*27 + 25*53 = 1460 ?

135 + 1325 = 1460 ?

1460 = 1460 ?yes

(2) substitution - isolate (solve) for one variable in terms of the other in one equation, then substitute that into the other equation

N = 80 - Q [from eq1]

5(80-Q) + 25Q = 1460 [substitute for N in eq2]

400 - 5Q + 25Q = 1460

400 + 20Q = 1460

20Q = 1060

Now, substitute for Q in either equation:

N + 53 = 80 [eq1 is pretty easy]

[note: same check applies]

*Let n be the number of nickels. *

*Let q be the number of quarters. *

We are told: q + n = 80 ,call this equation1

Also we know the dollar values: 0.25q + 0.05n = 14.60 ,call this equation2

Let's simplify equation1 and get, n = 80 - q

Let's plug n into equation2 and get, 0.25q + 0.05(80-q) = 14.6

Let's expand and get, 0.25q + 0.05*80 - 0.05*q =14.6

This simplifies to: 0.25q + 4 - 0.05q =14.6

Bring like terms together and get: 0.25q - 0.05q =14.6 - 4

This simplifies to: 0.20q = 10. 6

Solve for q and get, q = 10.6/0.20 = 53

From equation1 we know: q + n =80, Let's plug in q=53 and get, 53 + n = 80

Let's solve for n and get, n = 80 - 53 = 27

**Answer: So we have 27 nickels and 53 quarters. **

Extra step: Let's check our answer by using equation2:

0.25*53 +0.05*27 = 13.25+1.35 = 14.60

Our calculated 14.6 = 14.6 from equation2.

So we are correct!

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