Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Alice has a home-based business making and selling scented soaps. She intially spent $\$60$ to purchase soap-making equipment, and the materials for each pound of soap cost $\$2$ . Alice sells the soap for $\$12$ per pound. Eventually, she will sell enough soap to cover the cost of the equipment. What will be Alice's total sales and costs be? How much soap will that be? $\newline$ Alice's sales and costs will both be $\$\_\_\_\_$ once she sells $\_\_\_\_$ pounds of soap.

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Solve a system of equations using substitution: word problems

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

\newline

Alice has a home-based business making and selling scented soaps. She intially spent

\$60

to purchase soap-making equipment, and the materials for each pound of soap cost

\$2

. Alice sells the soap for

\$12

per pound. Eventually, she will sell enough soap to cover the cost of the equipment. What will be Alice's total sales and costs be? How much soap will that be?

\newline

Alice's sales and costs will both be

\$\_\_\_\_

once she sells

\_\_\_\_

pounds of soap.

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of pounds of soap Alice sells. $\newline$ Let $y$ be the total sales Alice makes from selling $x$ pounds of soap. $\newline$ Let $z$ be the total costs Alice incurs from making $x$ pounds of soap. $\newline$ Now, we can write two equations to represent the situation: $\newline$ $1$ . Total sales equation: $y = 12x$ (since Alice sells the soap for $\$12$ per pound) $\newline$ $2$ . Total costs equation: $z = 2x + 60$ (since each pound of soap costs $\$2$ to make and Alice initially spent $\$60$ on equipment) $\newline$ We want to find the point where total sales equal total costs, so we set $y$ equal to $z$ .
Write Equations: Now we substitute the expressions for y and z to find the value of x: $\newline$ $12x = 2x + 60$
Solve Equations: We solve for $x$ by subtracting $2x$ from both sides of the equation: $\newline$ $12x - 2x = 60$ $\newline$ $10x = 60$
Substitute and Calculate: Now we divide both sides by $10$ to find the value of $x$ : $x = \frac{60}{10}$ $x = 6$ This means Alice needs to sell $6$ pounds of soap to cover her costs.
Final Results: To find Alice's total sales and costs when she sells $6$ pounds of soap, we substitute $x$ back into the equations for $y$ and $z$ : $\newline$ Total sales ( $y$ ) = $12x = 12 \times 6 = \$(72)$ $\newline$ Total costs ( $z$ ) = $2x + 60 = 2 \times 6 + 60 = \$(72)$
Final Results: To find Alice's total sales and costs when she sells $6$ pounds of soap, we substitute $x$ back into the equations for $y$ and $z$ : $\newline$ Total sales ( $y$ ) = $12x = 12 \times 6 = \$(72)$ $\newline$ Total costs ( $z$ ) = $2x + 60 = 2 \times 6 + 60 = \$(72)$ We have found that Alice's sales and costs will both be $\$(72)$ once she sells $6$ pounds of soap. This answers the question prompt.