Answer :
Let's break down the problem step by step to find a solution for Carmen's packing task and her constraints.
### Step 1: Understand the Equations
We have the following system of equations representing Carmen's situation:
1. [tex]\(10x + 15y = 100\)[/tex]
- This equation represents the time constraint. Carmen can spend no more than 100 minutes packing boxes, where [tex]\(x\)[/tex] is the number of 4-pound treat boxes that take 10 minutes each to pack, and [tex]\(y\)[/tex] is the number of 2-pound game boxes that take 15 minutes each to pack.
2. [tex]\(4x + 2y = 50\)[/tex]
- This equation represents the weight constraint. Carmen can carry up to 50 pounds in total, where [tex]\(x\)[/tex] represents the number of 4-pound treat boxes, and [tex]\(y\)[/tex] represents the number of 2-pound game boxes.
### Step 2: Solve for Time Constraint
Let's solve the time equation for [tex]\(y\)[/tex]:
[tex]\[ 10x + 15y = 100 \][/tex]
First, divide each term by 5 to simplify:
[tex]\[ 2x + 3y = 20 \][/tex]
Rearrange to solve for [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
[tex]\[ 3y = 20 - 2x \][/tex]
[tex]\[ y = \frac{20 - 2x}{3} \][/tex]
### Step 3: Solve for Weight Constraint
Now, solve the weight equation for [tex]\(y\)[/tex]:
[tex]\[ 4x + 2y = 50 \][/tex]
First, divide each term by 2 to simplify:
[tex]\[ 2x + y = 25 \][/tex]
Rearrange to solve for [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
[tex]\[ y = 25 - 2x \][/tex]
### Step 4: Graph the Inequalities
To graph these inequalities, we can use the derived equations:
1. [tex]\( y = \frac{20 - 2x}{3} \)[/tex] from the time constraint.
2. [tex]\( y = 25 - 2x \)[/tex] from the weight constraint.
### Step 5: Find Intersection Point
To find where the two lines intersect, set the equations equal to each other:
[tex]\[ \frac{20 - 2x}{3} = 25 - 2x \][/tex]
Multiply through by 3 to clear the fraction:
[tex]\[ 20 - 2x = 75 - 6x \][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ 20 - 2x = 75 - 6x \][/tex]
[tex]\[ 4x = 55 \][/tex]
[tex]\[ x = \frac{55}{4} \][/tex]
[tex]\[ x = 13.75 \][/tex]
### Checking Feasibility
Let's check the feasibility of combining integers for [tex]\(x\)[/tex] and [tex]\(y\)[/tex], and most importantly, select those which satisfy both the inequalities and practical requirements. Given we require integer solutions, approximate reasonable solutions to guide Carmen’s packing.
[tex]\[ y = 25 - 2x \][/tex]
Substitute [tex]\( x = 5 \)[/tex] and so forth, ensuring all constraints and practicality.
With methodical checking, ensuring all values, treat boxes (x) and game boxes (y) are integers, reasonable solutions balancing all constraints can be computed for the exact number of each box.
Therefore, select correct intersections and feasible values via testing practical allowable solutions within constraints.
### Step 1: Understand the Equations
We have the following system of equations representing Carmen's situation:
1. [tex]\(10x + 15y = 100\)[/tex]
- This equation represents the time constraint. Carmen can spend no more than 100 minutes packing boxes, where [tex]\(x\)[/tex] is the number of 4-pound treat boxes that take 10 minutes each to pack, and [tex]\(y\)[/tex] is the number of 2-pound game boxes that take 15 minutes each to pack.
2. [tex]\(4x + 2y = 50\)[/tex]
- This equation represents the weight constraint. Carmen can carry up to 50 pounds in total, where [tex]\(x\)[/tex] represents the number of 4-pound treat boxes, and [tex]\(y\)[/tex] represents the number of 2-pound game boxes.
### Step 2: Solve for Time Constraint
Let's solve the time equation for [tex]\(y\)[/tex]:
[tex]\[ 10x + 15y = 100 \][/tex]
First, divide each term by 5 to simplify:
[tex]\[ 2x + 3y = 20 \][/tex]
Rearrange to solve for [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
[tex]\[ 3y = 20 - 2x \][/tex]
[tex]\[ y = \frac{20 - 2x}{3} \][/tex]
### Step 3: Solve for Weight Constraint
Now, solve the weight equation for [tex]\(y\)[/tex]:
[tex]\[ 4x + 2y = 50 \][/tex]
First, divide each term by 2 to simplify:
[tex]\[ 2x + y = 25 \][/tex]
Rearrange to solve for [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
[tex]\[ y = 25 - 2x \][/tex]
### Step 4: Graph the Inequalities
To graph these inequalities, we can use the derived equations:
1. [tex]\( y = \frac{20 - 2x}{3} \)[/tex] from the time constraint.
2. [tex]\( y = 25 - 2x \)[/tex] from the weight constraint.
### Step 5: Find Intersection Point
To find where the two lines intersect, set the equations equal to each other:
[tex]\[ \frac{20 - 2x}{3} = 25 - 2x \][/tex]
Multiply through by 3 to clear the fraction:
[tex]\[ 20 - 2x = 75 - 6x \][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ 20 - 2x = 75 - 6x \][/tex]
[tex]\[ 4x = 55 \][/tex]
[tex]\[ x = \frac{55}{4} \][/tex]
[tex]\[ x = 13.75 \][/tex]
### Checking Feasibility
Let's check the feasibility of combining integers for [tex]\(x\)[/tex] and [tex]\(y\)[/tex], and most importantly, select those which satisfy both the inequalities and practical requirements. Given we require integer solutions, approximate reasonable solutions to guide Carmen’s packing.
[tex]\[ y = 25 - 2x \][/tex]
Substitute [tex]\( x = 5 \)[/tex] and so forth, ensuring all constraints and practicality.
With methodical checking, ensuring all values, treat boxes (x) and game boxes (y) are integers, reasonable solutions balancing all constraints can be computed for the exact number of each box.
Therefore, select correct intersections and feasible values via testing practical allowable solutions within constraints.
