Math / Algebra

QuestionKitchen Korner produces refrigerators, dishwashers, and stoves at three different factories. The table gives the number of each product produced at each factory per day. Kitchen Korner receives an order for 166 refrigerators, 210 dishwashers, and 172 ovens. How many days should each plant be scheduled to fill this order? \begin{tabular}{|l|c|c|c|} \hline Appliance & Factory A & Factory B & Factory C \\ \hline Refrigerators & 8 & 10 & 14 \\ Dishwashers & 16 & 12 & 10 \\ Stoves & 10 & 18 & 6 \\ \hline \end{tabular}

Studdy Solution

STEP 1

1. Each factory produces a fixed number of each appliance per day.
2. The goal is to determine the number of days each factory should operate to meet the exact order requirements.
3. We can use a system of linear equations to solve the problem.

STEP 2

1. Define variables for the number of days each factory operates.
2. Set up equations based on the production rates and order requirements.
3. Solve the system of equations.

STEP 3

Define variables: Let $x$ be the number of days Factory A operates, $y$ be the number of days Factory B operates, and $z$ be the number of days Factory C operates.

STEP 4

Set up equations based on the production rates and order requirements:
For refrigerators: $8x + 10y + 14z = 166$
For dishwashers: $16x + 12y + 10z = 210$
For stoves: $10x + 18y + 6z = 172$

STEP 5

Solve the system of equations using substitution or elimination methods.
First, simplify the equations if possible. Let's start with the first two equations:
1. $8x + 10y + 14z = 166$
2. $16x + 12y + 10z = 210$
Multiply the first equation by 2 to align the coefficients of $x$ :
$16x + 20y + 28z = 332$
Subtract the second equation from this new equation:
$(16x + 20y + 28z) - (16x + 12y + 10z) = 332 - 210$
$8y + 18z = 122$

STEP 6

Now, solve for $y$ in terms of $z$ :
$8y = 122 - 18z$
$y = \frac{122 - 18z}{8}$
Substitute $y$ in terms of $z$ into the third equation:
$10x + 18\left(\frac{122 - 18z}{8}\right) + 6z = 172$
Simplify and solve for $x$ and $z$ :
$10x + \frac{2196 - 324z}{8} + 6z = 172$
Multiply through by 8 to clear the fraction:
$80x + 2196 - 324z + 48z = 1376$
$80x - 276z = 1376 - 2196$
$80x - 276z = -820$

STEP 7

Solve for $x$ in terms of $z$ :
$80x = 276z - 820$
$x = \frac{276z - 820}{80}$
Now, substitute both expressions for $x$ and $y$ in terms of $z$ into the first equation to find $z$ .
$8\left(\frac{276z - 820}{80}\right) + 10\left(\frac{122 - 18z}{8}\right) + 14z = 166$
Simplify and solve for $z$ :
$\frac{2208z - 6560}{80} + \frac{1220 - 180z}{8} + 14z = 166$
Clear fractions by multiplying through by 80:
$2208z - 6560 + 10(1220 - 180z) + 1120z = 13280$
$2208z - 6560 + 12200 - 1800z + 1120z = 13280$
Combine like terms:
$1528z + 5640 = 13280$
$1528z = 13280 - 5640$
$1528z = 7640$
$z = \frac{7640}{1528}$
$z = 5$

STEP 8

Substitute $z = 5$ back into the expressions for $x$ and $y$ :
$x = \frac{276(5) - 820}{80}$
$x = \frac{1380 - 820}{80}$
$x = \frac{560}{80}$
$x = 7$
$y = \frac{122 - 18(5)}{8}$
$y = \frac{122 - 90}{8}$
$y = \frac{32}{8}$
$y = 4$
The number of days each plant should be scheduled to fill the order is: Factory A: 7 days, Factory B: 4 days, Factory C: 5 days.

Was this helpful?

Studdy Solution

STEP 1

1. Each factory produces a fixed number of each appliance per day.
2. The goal is to determine the number of days each factory should operate to meet the exact order requirements.
3. We can use a system of linear equations to solve the problem.

STEP 2

1. Define variables for the number of days each factory operates.
2. Set up equations based on the production rates and order requirements.
3. Solve the system of equations.

STEP 3

Define variables: Let $x$ be the number of days Factory A operates, $y$ be the number of days Factory B operates, and $z$ be the number of days Factory C operates.

STEP 4

Set up equations based on the production rates and order requirements:
For refrigerators: $8x + 10y + 14z = 166$
For dishwashers: $16x + 12y + 10z = 210$
For stoves: $10x + 18y + 6z = 172$

STEP 5

STEP 6

STEP 7

STEP 8

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

1. Each factory produces a fixed number of each appliance per day.2. The goal is to determine the number of days each factory should operate to meet the exact order requirements.3. We can use a system of linear equations to solve the problem.

STEP 2

1. Define variables for the number of days each factory operates.2. Set up equations based on the production rates and order requirements.3. Solve the system of equations.

STEP 3

Define variables: Let x x x be the number of days Factory A operates, y y y be the number of days Factory B operates, and z z z be the number of days Factory C operates.

STEP 4

Set up equations based on the production rates and order requirements:For refrigerators: 8x+10y+14z=166 8x + 10y + 14z = 166 8x+10y+14z=166For dishwashers: 16x+12y+10z=210 16x + 12y + 10z = 210 16x+12y+10z=210For stoves: 10x+18y+6z=172 10x + 18y + 6z = 172 10x+18y+6z=172

STEP 5

STEP 6

STEP 7

STEP 8

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Each factory produces a fixed number of each appliance per day.
2. The goal is to determine the number of days each factory should operate to meet the exact order requirements.
3. We can use a system of linear equations to solve the problem.

1. Define variables for the number of days each factory operates.
2. Set up equations based on the production rates and order requirements.
3. Solve the system of equations.

Define variables: Let $x$ be the number of days Factory A operates, $y$ be the number of days Factory B operates, and $z$ be the number of days Factory C operates.

Set up equations based on the production rates and order requirements:
For refrigerators: $8x + 10y + 14z = 166$
For dishwashers: $16x + 12y + 10z = 210$
For stoves: $10x + 18y + 6z = 172$