number of cups of lemonade --- x
number of cupcakes ---- y
2x + y ≥ 25, where x > 5, y > 5
x .. y
5 15
6 13
7 11
...
How Do I solve this system of linear equalities word problem?
"Sandy makes $2 profit on every cup of lemonade that she sells and $1 on every cupcake that she sells. Sandy wants to sell at least 5 cups of lemonade and at least 5 cupcakes per day. She wants to earn at least $25 per day. Show and describe all the possible combinations of lemonade and cupcakes that Sandy needs to sell to meet her goals. List two possible combinations."
3 answers
Let
x = # cups of lemonade
y = # cupcakes
What have they told you?
x >= 5
y >= 5
2x + 1y >= 25
So, start with the smallest value of x, and then let it grow.
Start with the smallest acceptable profit: 25
x y profit
5 15 25
6 13 25
...
Now, raise the profit to 26 and do it again
5 16 26
6 14 26
...
Or, graph all three of those lines, and shade the solution sets.
Then just pick any points in the shaded area.
x = # cups of lemonade
y = # cupcakes
What have they told you?
x >= 5
y >= 5
2x + 1y >= 25
So, start with the smallest value of x, and then let it grow.
Start with the smallest acceptable profit: 25
x y profit
5 15 25
6 13 25
...
Now, raise the profit to 26 and do it again
5 16 26
6 14 26
...
Or, graph all three of those lines, and shade the solution sets.
Then just pick any points in the shaded area.
C = cupcakes
p = profit
Sandy makes $2 profit on every cup of lemonade that she sells and $1 on every cupcake that she sells neans:
p = $2 L + $1C
p = 2 L + C
She wants to earn at least $25 per day means:
p ≥ 25
2 L + C ≥ 25
Sandy wants to sell at least 5 cups of lemonade and at least 5 cupcakes per day means:
L ≥ 5
C ≥ 5
Start with L = 5
2 L + C ≥ 25
2 ∙ 5 + C ≥ 25
10 + C ≥ 25
C ≥ 15
L = 6
2 L + C ≥ 25
2 ∙ 6 + C ≥ 25
12 + C ≥ 25
C ≥ 13
L = 7
2 L + C ≥ 25
2 ∙ 7 + C ≥ 25
14 + C ≥ 25
C ≥ 11
L = 8
2 L + C ≥ 25
2 ∙ 8 + C ≥ 25
16 + C ≥ 25
C ≥ 9
L = 9
2 L + C ≥ 25
2 ∙ 9 + C ≥ 25
18 + C ≥ 25
C ≥ 7
L = 10
2 L + C ≥ 25
2 ∙ 10 + C ≥ 25
20 + C ≥ 25
C ≥ 5
L = 11
2 L + C ≥ 25
2 ∙ 11 + C ≥ 25
22 + C ≥ 25
C ≥ 3
L = 12
2 L + C ≥ 25
2 ∙ 12 + C ≥ 25
24 + C ≥ 25
C ≥ 1
For L = 13
2 L + C ≥ 25
2 ∙ 13 + C ≥ 25
26 + C ≥ 25
C ≥ - 1
For L ≥ 13 , C is negative so you must reject cobinations L ≥ 13 C. In other words, for L ≥ 13 the lemonade sold would earn more than $25, so she wouldn't even sell cupcakes.
So possible combinations are:
L = 5 , C ≥ 15
L = 6 , C ≥ 13
L = 7 , C ≥ 11
L = 8 , C ≥ 9
L = 9 , C ≥ 7
L = 10 , C ≥ 5
L = 11 , C ≥ 3
L = 12 , C ≥ 1
This is the MINIMUM number of combinations that earns the required profit.
Of course she can sell more than this.
p = profit
Sandy makes $2 profit on every cup of lemonade that she sells and $1 on every cupcake that she sells neans:
p = $2 L + $1C
p = 2 L + C
She wants to earn at least $25 per day means:
p ≥ 25
2 L + C ≥ 25
Sandy wants to sell at least 5 cups of lemonade and at least 5 cupcakes per day means:
L ≥ 5
C ≥ 5
Start with L = 5
2 L + C ≥ 25
2 ∙ 5 + C ≥ 25
10 + C ≥ 25
C ≥ 15
L = 6
2 L + C ≥ 25
2 ∙ 6 + C ≥ 25
12 + C ≥ 25
C ≥ 13
L = 7
2 L + C ≥ 25
2 ∙ 7 + C ≥ 25
14 + C ≥ 25
C ≥ 11
L = 8
2 L + C ≥ 25
2 ∙ 8 + C ≥ 25
16 + C ≥ 25
C ≥ 9
L = 9
2 L + C ≥ 25
2 ∙ 9 + C ≥ 25
18 + C ≥ 25
C ≥ 7
L = 10
2 L + C ≥ 25
2 ∙ 10 + C ≥ 25
20 + C ≥ 25
C ≥ 5
L = 11
2 L + C ≥ 25
2 ∙ 11 + C ≥ 25
22 + C ≥ 25
C ≥ 3
L = 12
2 L + C ≥ 25
2 ∙ 12 + C ≥ 25
24 + C ≥ 25
C ≥ 1
For L = 13
2 L + C ≥ 25
2 ∙ 13 + C ≥ 25
26 + C ≥ 25
C ≥ - 1
For L ≥ 13 , C is negative so you must reject cobinations L ≥ 13 C. In other words, for L ≥ 13 the lemonade sold would earn more than $25, so she wouldn't even sell cupcakes.
So possible combinations are:
L = 5 , C ≥ 15
L = 6 , C ≥ 13
L = 7 , C ≥ 11
L = 8 , C ≥ 9
L = 9 , C ≥ 7
L = 10 , C ≥ 5
L = 11 , C ≥ 3
L = 12 , C ≥ 1
This is the MINIMUM number of combinations that earns the required profit.
Of course she can sell more than this.