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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineNicole has a home-based business making and selling scented soaps. She intially spent $55\$55 to purchase soap-making equipment, and the materials for each pound of soap cost $4\$4. Nicole sells the soap for $15\$15 per pound. Eventually, she will sell enough soap to cover the cost of the equipment. How much soap will that be? What will be Nicole's total sales and costs be?\newlineOnce Nicole sells _____ pounds of soap, her sales and costs will both be $\$_____.

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineNicole has a home-based business making and selling scented soaps. She intially spent $55\$55 to purchase soap-making equipment, and the materials for each pound of soap cost $4\$4. Nicole sells the soap for $15\$15 per pound. Eventually, she will sell enough soap to cover the cost of the equipment. How much soap will that be? What will be Nicole's total sales and costs be?\newlineOnce Nicole sells _____ pounds of soap, her sales and costs will both be $\$_____.
  1. Define Variables and Equations: Define the variables and write the equations.\newlineLet xx be the number of pounds of soap Nicole sells.\newlineNicole's total cost (CC) includes the initial equipment cost and the cost of materials per pound of soap.\newlineC=initial equipment cost+(cost per pound of soap×x)C = \text{initial equipment cost} + (\text{cost per pound of soap} \times x)\newlineC=$(55)+$(4)xC = \$(55) + \$(4)x\newlineNicole's total sales (SS) are based on the selling price per pound of soap.\newlineS=selling price per pound of soap×xS = \text{selling price per pound of soap} \times x\newlineS=$(15)xS = \$(15)x
  2. Set Break-Even Equation: Set up the equation to find the break-even point where sales equal costs.\newline15x=55+4x15x = 55 + 4x
  3. Solve for Break-Even Point: Solve for xx using substitution or combination.\newline$15x$4x=$55\$15x - \$4x = \$55\newline$11x=$55\$11x = \$55\newlinex=$55/11x = \$55 / 11\newlinex=5x = 5
  4. Calculate Sales and Costs: Calculate the total sales and costs when Nicole sells 55 pounds of soap.\newlineTotal sales (S) when x=5x = 5:\newlineS=$(15)×5S = \$(15) \times 5\newlineS=$75S = \$75\newlineTotal costs (C) when x=5x = 5:\newlineC=$(55)+$(4)×5C = \$(55) + \$(4) \times 5\newlineC=$(55)+$(20)C = \$(55) + \$(20)\newlineC=$75C = \$75
  5. Fill in Solution: Fill in the blanks with the solution.\newlineOnce Nicole sells 55 pounds of soap, her sales and costs will both be $75\$75.

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