Question

The drama club is selling short-sleeved shirts for $5 each, and long-sleeved shirts for $10 each. They hope to sell all of the shirts they ordered, to earn a total of $1,750. After the first week of the fundraiser, they sold  1/3 of the short-sleeved shirts and  1/2of the long-sleeved shirts, for a total of 100 shirts.

This system of equations models the situation.
   5x + 10y = 1,750
1/3x + 1/2y = 100
Let x represent the number of short-sleeved shirts ordered and let y represent the number of long-sleeved shirts ordered.
How many short-sleeved shirts were ordered?

How many long-sleeved shirts were ordered?

Answer 1

150 short-sleeved shirts
100 long-sleeved shirts

Answer 2

150 short-sleeved shirts and 100 long-sleeved shirts.

Explanation:

We will use elimination to solve this system of equations. We will make the coefficients of x the same in order to eliminate them; to do this, we must multiply the bottom equation by 15 (1/3 of 15 is 5, so 15(1/3) = 5):

`\left \{ {{5x+10y=1750} \atop {15((1)/(3)x+(1)/(2)y=100)} \right. \\\\\left \{ {{5x+10y=1750} \atop {5x+7.5y=1500}} \right.`

Subtract the second equation from the first one:

`\left \{ {{5x+10y=1750} \atop {-(5x+7.5y=1500)}} \right. \\\\2.5y=250`

Divide both sides by 2.5:

2.5y/2.5 = 250/2.5

y = 100

Substitute 100 in for y in the first equation:

5x+10(100) = 1750

5x+1000 = 1750

Subtract 1000 from both sides:

5x+1000 - 1000 = 1750 - 1000

5x = 750

Divide both sides by 5:

5x/5 = 750/5

x = 150