Question 10389
Let's decipher the first sentence: "Brand X bottle contains 1 dozen fewer capsules than the Brand Y bottle". This is saying that brand X has less capsules in it than brand Y. It is VERY important to FIRST DETERMINE which one has more content. In this case, the Y contains more. If Y contains more, then it's the bottle that has to be knocked down by 12 capsules to equal the (fewer) number of capsules in bottle X. Our equation then is {{{ X = Y - 12 }}}.


They then told us that 5 brand X bottles and 8 brand Y bottles have a total of 486 capsules. SO, 5 bottles * X capsules per bottle would give us the number of X-type capsules. Also, 8 bottles * Y capsules per bottle would give us the number of Y-type capsules. We need to total up the number of X and Y capsules to equal 486: {{{ 5X + 8Y = 486 }}}.


Since we know about how many capsules of X there are in one bottle compared to how many capsules of Y there are in that bottle, we need to make a substitution. Since {{{ X = Y - 12 }}}, we can substitute {{{ Y-12 }}} for the X in the equation {{{ 5X + 8Y = 486 }}}.


{{{ 5(Y-12) + 8Y = 486 }}} <------- Notice that we placed parentheses around the Y-12. It's ALWAYS a good practice to place parentheses around what you substituted. We substituted so that the equation will only be in 1 variable. This way, you can solve the equation.


{{{ 5Y - 60 + 8Y = 486 }}} <----- distributed


{{{ 13Y - 60 = 486 }}} <-------- combine like terms


{{{ 13Y = 546 }}} <----- Add 60 to both sides


{{{ Y = 42 }}} <------ There are 42 capsules in the Y-brand bottle. The problem does ask for how many X-capsules are in one bottle. Since We know that there are 12 less X-capsules in one bottle than there are Y-capsules, then there would be 30 capsules in the X-bottle. We actually plugged in the 42 in the equation {{{ X = Y - 12 }}} to get the x. {{{ 30 = 42 - 12 }}}.