Question

6-46. mixing candy
renard thought that writing two equations for problem 6-44 was easy. he wants to use two equations with two variables to solve this problem:
ariel bought several bags of caramel candy and several bags of taffy. the number of bags of taffy was 5 more than the number of bags of caramels. taffy bags weigh 8 ounces each, and caramel bags weigh 16 ounces each. the total weight of all the bags of candy was 400 ounces. how many bags of candy did she buy?
a. renard lets $t=$ the number of taffy bags and $c=$ the number of caramel bags. help him write two equations to represent the information in the problem.
b. now renard is stuck. he says, \if both of the equations were in the form t= something, i could set the two expressions equal to each other to find the solution.\ help him rewrite the equations into a form he can use to solve the problem.

Explanation:

Step1: Define equations for part a

First equation (count relation): $t = c + 5$
Second equation (total weight): $8t + 16c = 400$

Step2: Rewrite second equation for part b

Isolate $t$ from weight equation.
Divide all terms by 8:
$\frac{8t}{8} + \frac{16c}{8} = \frac{400}{8}$
Simplify to get: $t = 50 - 2c$
Keep first equation as is: $t = c + 5$

Answer:

a. The two equations are:
$t = c + 5$
$8t + 16c = 400$

b. The rewritten equations in $t=$ form are:
$t = c + 5$
$t = 50 - 2c$