Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Malone's Bakery sold one customer $1$ dozen chocolate cookies and $10$ dozen oatmeal cookies for $\$76$ . The bakery also sold another customer $10$ dozen chocolate cookies and $5$ dozen oatmeal cookies for $\$95$ . 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 elimination: word problems

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

\newline

Malone's Bakery sold one customer

1

dozen chocolate cookies and

10

dozen oatmeal cookies for

\$76

. The bakery also sold another customer

10

dozen chocolate cookies and

5

dozen oatmeal cookies for

\$95

. How much do the cookies cost?

\newline

A dozen chocolate cookies cost

\$

_____, and a dozen oatmeal cookies cost

\$

_____.

Define variables: Define the variables for the cost of a dozen chocolate cookies and a dozen oatmeal cookies. $\newline$ Let $x$ be the cost of a dozen chocolate cookies and $y$ be the cost of a dozen oatmeal cookies.
Write first equation: Write the equation for the first customer's purchase. $\newline$ $1$ dozen chocolate cookies and $10$ dozen oatmeal cookies cost $\$76$ . $\newline$ $1 \times x + 10 \times y = 76$ $\newline$ $x + 10y = 76$
Write second equation: Write the equation for the second customer's purchase. $\newline$ $10$ dozen chocolate cookies and $5$ dozen oatmeal cookies cost $\$95$ . $\newline$ $10 \times x + 5 \times y = 95$ $\newline$ $10x + 5y = 95$
Eliminate variable $x$ : Choose which variable to eliminate. $\newline$ We will eliminate $x$ by multiplying the first equation by $-10$ and the second equation by $1$ to make the coefficients of $x$ opposite. $\newline$ $-10(x + 10y) = -10(76)$ $\newline$ $10x + 5y = 95$
Multiply first equation: Multiply the first equation by -10＂).$\newline\$-10x - 100y = -760$
Add equations: Add the modified first equation to the second equation to eliminate $x$.\[(-10x - 100y) + (10x + 5y) = -760 + 95\]\[-10x + 10x - 100y + 5y = -760 + 95\]\[0x - 95y = -665\]\[-95y = -665\]
Solve for y: Solve for y.$\newline$Divide both sides by $-95$ to find the cost of a dozen oatmeal cookies.$\newline$$y = -665 / -95$$\newline$$y = 7$
Substitute for $x$: Substitute the value of $y$ into the first equation to solve for $x$.
$x + 10(7) = 76$
$x + 70 = 76$
$x = 76 - 70$
$x = 6$
Verify solution: Verify the solution by substituting $x$ and $y$ into the second equation.\[10(6) + 5(7) = 95\]\[60 + 35 = 95\]\[95 = 95\]The solution is verified.