Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Livingston's Bakery sold one customer $8$ dozen chocolate cookies and $10$ dozen oatmeal cookies for $\$104$ . The bakery also sold another customer $4$ dozen chocolate cookies and $3$ dozen oatmeal cookies for $\$44$ . How much do the cookies cost? $\newline$ A dozen chocolate cookies cost $\$\_\_\_\_$ , and a dozen oatmeal cookies cost $\$\_\_\_\_$ .

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Algebra 1

Solve a system of equations using any method: word problems

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

Livingston's Bakery sold one customer

8

dozen chocolate cookies and

10

dozen oatmeal cookies for

\$104

. The bakery also sold another customer

4

dozen chocolate cookies and

3

dozen oatmeal cookies for

\$44

. How much do the cookies cost?

\newline

A dozen chocolate cookies cost

\$\_\_\_\_

, and a dozen oatmeal cookies cost

\$\_\_\_\_

Define Cookie Costs: Let's denote the cost of a dozen chocolate cookies as $c$ dollars and the cost of a dozen oatmeal cookies as $o$ dollars. $\newline$ The first customer bought $8$ dozen chocolate cookies and $10$ dozen oatmeal cookies for $104$. This can be represented by the equation:$\newline$$ 8c + 10o = 104 $
First Customer Purchase: The second customer bought $4$ dozen chocolate cookies and $3$ dozen oatmeal cookies for $44$ . This can be represented by the equation: $\newline$ $4c + 3o = 44$
Second Customer Purchase: We now have a system of equations to solve: $\newline$ $8c + 10o = 104$ $\newline$ $4c + 3o = 44$ $\newline$ We can use either substitution or elimination to solve this system. Let's use the elimination method.
Solve System of Equations: To eliminate one of the variables, we can multiply the second equation by $2$ to make the coefficient of $c$ the same in both equations: $\newline$ $2(4c + 3o) = 2(44)$ $\newline$ $8c + 6o = 88$
Eliminate Variable c: Now we subtract the new equation from the first equation to eliminate $c$ : $\newline$ $(8c + 10o) - (8c + 6o) = 104 - 88$ $\newline$ $4o = 16$
Find Cost of Oatmeal Cookies: Solving for $o$ , we get: $\newline$ $o = \frac{16}{4}$ $\newline$ $o = 4$ $\newline$ So, a dozen oatmeal cookies cost $4$.
Substitute to Find Cost of Chocolate Cookies: Now we can substitute the value of $ o $ into one of the original equations to find $ c $. Let's use the second equation:$\newline$$ 4c + 3(4) = 44 $$\newline$$ 4c + 12 = 44 $$\newline$$ 4c = 44 - 12 $$\newline$$ 4c = 32 $
Substitute to Find Cost of Chocolate Cookies: Now we can substitute the value of $ o $ into one of the original equations to find $ c $. Let's use the second equation:$\newline$$ 4c + 3(4) = 44 $$\newline$$ 4c + 12 = 44 $$\newline$$ 4c = 44 - 12 $$\newline$$ 4c = 32 $Solving for $ c $, we get:$\newline$$ c = \frac{32}{4} $$\newline$$ c = 8 $$\newline$So, a dozen chocolate cookies cost $8$ .