Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ The Hillsboro Tigers are having team shirts made. One option is to pay Clarence's Tees a $\$ 50$ setup fee and then buy the shirts for $\$ 5$ each. Another option is to go to City Printing, paying $\$ 36$ for a setup fee and an additional $\$ 12$ per shirt. The team parent in charge of the project notices that, with a certain number of shirts, the two options have the same cost. How many shirts is that? What is the cost? $\newline$ If the Tigers have shirts made, the cost is $\$ \square$ . $\newline$ Submit

View step-by-step help

Home

Math Problems

Algebra 2

Solve a system of equations using substitution: word problems

Full solution

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

\newline

The Hillsboro Tigers are having team shirts made. One option is to pay Clarence's Tees a

\$ 50

setup fee and then buy the shirts for

\$ 5

each. Another option is to go to City Printing, paying

\$ 36

for a setup fee and an additional

\$ 12

per shirt. The team parent in charge of the project notices that, with a certain number of shirts, the two options have the same cost. How many shirts is that? What is the cost?

\newline

If the Tigers have shirts made, the cost is

\$ \square

\newline

Submit

Define Variables and Equations: Define the variables and write the system of equations. $\newline$ Let $x$ represent the number of shirts and $y$ represent the total cost. $\newline$ For Clarence's Tees: $\newline$ Setup fee: $\$50$ $\newline$ Cost per shirt: $\$5$ $\newline$ Equation: Total cost $y$ = setup fee + (cost per shirt * number of shirts) $\newline$ $y = 5x + 50$ $\newline$ For City Printing: $\newline$ Setup fee: $\$36$ $\newline$ Cost per shirt: $\$12$ $\newline$ Equation: Total cost $y$ = setup fee + (cost per shirt * number of shirts) $\newline$ $y = 12x + 36$ $\newline$ System of equations: $\newline$ $y = 5x + 50$ $\newline$ $y = 12x + 36$
Substitution and Solving: Solve the system using substitution. $\newline$ Since both equations equal $y$ , set them equal to each other to find the number of shirts $x$ where the cost is the same. $\newline$ $5x + 50 = 12x + 36$
Isolate Variable x: Isolate the variable x. $\newline$ Subtract $5x$ from both sides of the equation: $\newline$ $5x + 50 - 5x = 12x + 36 - 5x$ $\newline$ $50 = 7x + 36$ $\newline$ Now, subtract $36$ from both sides: $\newline$ $50 - 36 = 7x + 36 - 36$ $\newline$ $14 = 7x$
Solve for x: Solve for x. $\newline$ Divide both sides by $7$ to find the value of $x$ : $\newline$ $\frac{14}{7} = \frac{7x}{7}$ $\newline$ $2 = x$ $\newline$ So, $x = 2$ , which means the same cost occurs when $2$ shirts are made.
Find Total Cost: Find the total cost $y$ when $x = 2$ . $\newline$ Substitute $x = 2$ into one of the original equations to find $y$ . We'll use the equation from Clarence's Tees: $\newline$ $y = 5(2) + 50$ $\newline$ $y = 10 + 50$ $\newline$ $y = 60$ $\newline$ So, the total cost $y$ is $\$60$ when $2$ shirts are made.