Answer:
A = 100,
B = 87.
And the solution to the system us
[tex]x = 2[/tex], [tex]y =5.[/tex]
Step-by-step explanation:
Let us call [tex]x[/tex] the servings of grapes and [tex]y[/tex] he servings of raspberries.
One serving of grapes has 100 calories and one serving of raspberries has 60 calories, and since together 500 calories must be consumed, we have:
[tex]100x+60y=500[/tex]
Furthermore, one serving of grapes has 21 grams of carbohydrates and one serving of raspberries has 9 grams, and if at total of 87 grams of carbohydrates are consumed, we have
[tex]21x+9y=21[/tex].
Thus, the system of equations we have are
(1). [tex]100x+60y=500[/tex]
(2). [tex]21x+9y=87[/tex].
Here we see that
[tex]A = 100[/tex]
[tex]B = 87[/tex].
The numbers of servings that must be consumed are the solutions of the system.
The solutions to the system can be found by solving for [tex]x[/tex] in equation(2), and substituting its value equation (1):
[tex]x = \dfrac{87-9y}{21}[/tex]
[tex]100(\dfrac{87-9y}{21})+60y=500[/tex]
[tex]$\frac{2900}{7}-\frac{300y}{7} +60y=500 $[/tex]
[tex]$\frac{120}{7}y=\frac{600}{7} $[/tex]
[tex]\boxed{y=5}[/tex]
putting this into equation (2), we solve for [tex]x[/tex]:
[tex]21x+9*5=87\\21x+45=87\\21x=42\\\\\boxed{x = 2}[/tex]
