type the correct answer in each box.\nsusans business makes handmade dresses. the business sells three…

type the correct answer in each box.\nsusans business makes handmade dresses. the business sells three different kinds of dresses: evening gowns, sundresses, and cocktail dresses. to make an evening gown, it takes 3 hours to cut the material, 2 hours to pin the pieces together, and 5 hours to sew the dress. to make a sundress, it takes 1 hour to cut the material, 1.5 hours to pin the pieces together, and 3 hours to sew the dress. to make a cocktail dress, it takes 2 hours to cut the material, 1 hour to pin the pieces together, and 2.5 hours to sew the dress. susans employees spent a total of 54 hours cutting, 38.5 hours pinning, and 91 hours sewing in a particular week. how many of each type of dress did they produce?\nthat week, the employees produced evening gowns, sundresses, and cocktail dresses.

type the correct answer in each box.\nsusans business makes handmade dresses. the business sells three different kinds of dresses: evening gowns, sundresses, and cocktail dresses. to make an evening gown, it takes 3 hours to cut the material, 2 hours to pin the pieces together, and 5 hours to sew the dress. to make a sundress, it takes 1 hour to cut the material, 1.5 hours to pin the pieces together, and 3 hours to sew the dress. to make a cocktail dress, it takes 2 hours to cut the material, 1 hour to pin the pieces together, and 2.5 hours to sew the dress. susans employees spent a total of 54 hours cutting, 38.5 hours pinning, and 91 hours sewing in a particular week. how many of each type of dress did they produce?\nthat week, the employees produced evening gowns, sundresses, and cocktail dresses.

Answer

Explanation:

Step1: Set up equations

Let $x$ be the number of evening gowns, $y$ be the number of sundresses, and $z$ be the number of cocktail - dresses. The cutting - time equation is $3x + y+2z = 54$. The pinning - time equation is $2x + 1.5y+z = 38.5$. The sewing - time equation is $5x + 3y+2.5z = 91$.

Step2: Multiply the second equation by 2

Multiply $2x + 1.5y+z = 38.5$ by 2 to get $4x + 3y+2z = 77$.

Step3: Subtract the new equation from the sewing - time equation

$(5x + 3y+2.5z)-(4x + 3y+2z)=91 - 77$. $5x + 3y+2.5z-4x - 3y - 2z=14$. $x+0.5z = 14$, so $x = 14 - 0.5z$.

Step4: Substitute $x$ into the cutting - time equation

$3(14 - 0.5z)+y + 2z=54$. $42-1.5z+y + 2z=54$. $y+0.5z=12$, so $y = 12 - 0.5z$.

Step5: Substitute $x$ and $y$ into the second original equation

$2(14 - 0.5z)+1.5(12 - 0.5z)+z = 38.5$. $28 - z+18 - 0.75z+z = 38.5$. $28+18-(z + 0.75z - z)=38.5$. $46 - 0.75z=38.5$. $-0.75z=38.5 - 46=-7.5$. $z = 10$.

Step6: Find $x$ and $y$

Since $z = 10$, then $x = 14-0.5\times10=14 - 5 = 9$. $y = 12-0.5\times10=12 - 5 = 7$.

Answer:

9 evening gowns, 7 sundresses, and 10 cocktail dresses.