Candy Store Word Problem. Please help explain one part.

Question

Doug and Steve went to the candy store. Doug bought 5 pieces of fudge and 3 pieces
of bubble gum for a total of $5.70. Steve bought 2 pieces of fudge and 10 pieces of
bubble gum for a total of $3.60. How much does 1 piece of fudge and 1 piece of
bubble gum cost?

 

I already know the answer, but I'm trying to understand the logic behind the step up. I understand that I have to make an equation for each statement that looks like:

 

-2(5x +3y = 5.70)

5(2x +10y=3.60)

 

My question is why am I adding the -2 & 5 in front of the equation? Where does the logic come from? Am I doing it simply to cancel and isolate the y variable? For other similar questions, would I just always add the leading coefficient in negative form in front of the opposite equation? 

 

I hope that is clear! Thank you

Answer 1

They are not really there as part of the setup. They are more like the first step you would do to use the elimination method.

 

If you multiply top by -2 and bottom eq. by 5 and add the two equations. This would Eliminate the variable x and then you could easily solve for y.

Answer 2

Doug and Steve went to the candy store. Doug bought 5 pieces of fudge and 3 pieces
of bubble gum for a total of $5.70. Steve bought 2 pieces of fudge and 10 pieces of
bubble gum for a total of $3.60

5f + 3b = 5.70  That's Doug

2f + 10b = 3.60  That's Steve

 

We will solve simultaneously by either solving one equation for f (or b) in terms of the other variable, and then using substitution to solve the second equation for just one variable, or...

make either f or b equal in coefficients so that by subtraction, we can eliminate one.

Multiply this one by 2: 5f + 3b = 5.70

Multiply this one by 5: 2f + 10b = 3.60

You'll get new equations in which 10 is the coefficient of f.  So, by subtraction of one equation from another, you'll have an equation in only b, which you can then solve.  Then, go back and use one equation to solve for f, now that you know b.  Whew!