Aaliyah gets paid at home for doing extra chores. Last week, she did 6 loads of laundry and 5 loads of dishes, and her parents paid her $23. The week before, she finished 3 loads of laundry and 1 load of dishes, earning a total of $10. How much does Aaliyah earn for completing each type of chore?

Question

Aaliyah earns $
per load of laundry and $
per load of dishes.

Answer 1

Let's define the variables for Aaliyah's earnings:

• Let \( x \) be the amount she earns for each load of laundry.
• Let \( y \) be the amount she earns for each load of dishes.

We can set up the following equations based on the information given:

From the first week, where Aaliyah did 6 loads of laundry and 5 loads of dishes and earned $23:

\[ 6x + 5y = 23 \quad \text{(1)} \]

From the previous week, where she did 3 loads of laundry and 1 load of dishes and earned $10:

\[ 3x + 1y = 10 \quad \text{(2)} \]

Now we have a system of two equations with two variables. We can solve it using substitution or elimination. We'll use the substitution method.

First, let's solve equation (2) for \( y \):

\[ y = 10 - 3x \quad \text{(3)} \]

Now, we'll substitute equation (3) into equation (1):

\[ 6x + 5(10 - 3x) = 23 \]

Distributing the 5 in the equation:

\[ 6x + 50 - 15x = 23 \]

Combining like terms:

\[ -9x + 50 = 23 \]

Now isolate \( x \):

\[ -9x = 23 - 50 \] \[ -9x = -27 \] \[ x = 3 \]

Now that we found \( x \), we can substitute it back into equation (3) to find \( y \):

\[ y = 10 - 3(3) \] \[ y = 10 - 9 \] \[ y = 1 \]

Thus, Aaliyah earns:

• $3 per load of laundry
• $1 per load of dishes

In summary, Aaliyah earns $3 per load of laundry and $1 per load of dishes.

Answer 2

Dr. Duncan, a pediatrician, has 2 annual checkups and 3 sick visits scheduled next Tuesday, which will fill a total of 174 minutes on his schedule. Next Wednesday, he has 2 annual checkups and 1 sick visit on the schedule, which should take 122 minutes. How much time is allotted for each type of appointment?

The time allotted is
minutes for an annual checkup and
minutes for a sick visit.

Answer 3

Let's define the variables:

• Let \( a \) be the time allotted for each annual checkup (in minutes).
• Let \( s \) be the time allotted for each sick visit (in minutes).

We can set up the following equations based on the information given:

From the first day (Tuesday) where Dr. Duncan has 2 annual checkups and 3 sick visits scheduled for a total of 174 minutes:

\[ 2a + 3s = 174 \quad \text{(1)} \]

From the second day (Wednesday) where he has 2 annual checkups and 1 sick visit scheduled for a total of 122 minutes:

\[ 2a + 1s = 122 \quad \text{(2)} \]

Now we have a system of two equations with two variables. We can solve it using elimination or substitution. Here, we will use elimination.

First, we can simplify both equations. We can subtract equation (2) from equation (1):

Subtracting equation (2) from equation (1):

\[ (2a + 3s) - (2a + 1s) = 174 - 122 \]

This simplifies to:

\[ 2s = 52 \]

Dividing both sides by 2:

\[ s = 26 \]

Now we can substitute the value of \( s \) back into either equation (1) or (2) to find \( a \). We'll use equation (2):

\[ 2a + 1(26) = 122 \]

This simplifies to:

\[ 2a + 26 = 122 \]

Now isolate \( 2a \):

\[ 2a = 122 - 26 \] \[ 2a = 96 \] \[ a = 48 \]

Thus, the time allotted is:

• 48 minutes for an annual checkup
• 26 minutes for a sick visit

In summary, the time allotted is 48 minutes for an annual checkup and 26 minutes for a sick visit.