Answered

Assign two variables for each problem, and write the equations. (I do need these solved!)


1) At a toy store, the children's department has bicycles and tricycles. There are 50 total, and 111 wheels. How many bicycles are there?


2) Anna has a pocket of dimes and quarters. If she has 10 coins worth $1.45, how many of her coins are quarters?


Two variables, two equations, and I need them solved.

Verified Answer :

1)
lets make b equal to the number of bicycles and t equal to the number of tricycles.  since we know the total number of both, we can make the equation b+t=50.  Since we know the total number of wheels is 111 and that each bicycle has 2 wheels and each tricycle has 3 wheels, you can make the equation 2b+3t=111.  Since the b and t variable are the same you can make a system of equation as fallows:
b+t=50
2b+3t=111

Since you are tying to find the number of bicycles you need to some how cancel out the variable for tricycles.  The eaiest thing to in this case is multiply the top equation by -3 to get:
-3b-3t=-150  (you add the bottom function to the top function)
2b+3t=111     (-3b+2b=-b, -3t+3t=0, and -150+111=-39)
the -3t and 3t cancel out and you are left with the single equation of -b=-39 so the total number of bikes is 39.

2)
lets make d equal the number of dimes and q equal the number of quarters.  you know that she has 10 coins, therefore d+q=10.  You also know that the total amount equals 1.45  (which can be written as 145 since there are 100 cents in a dollar) and that each quarter is worth 25 and each dime is worth 10 which means you can write 10d+25q=1.45.  Since the q and t values are the same for both equations you can make a system of equations as follows:
d+q=10
10d+25q=145

since you are looking for the number of quarters, you need to somehow cancel out d which can be done by multiplying the top equation by -10.
-10d-10q=-100 (like before, add the top function to the bottom function)
10d+25q=145  (-10d+10d=0, -10q+25q=15q, and -100+145=45)
 since the d values cancel out, you are left with the single equation 15q=45 which means q=3.
The total number of quarters is 3.

I hope this helps.  Let me know in the comments if anything is unclear.

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