Algebra homework help

Question

Question: Souvenir hats, T-shirts, and jackets are sold at Bob land. All aliens and non-aliens are granted admission, given they can recite the quadratic formula at the ticket counter. Three hats, two T-shirts, and one jacket cost $140. Two hats, two T-shirts, and two jackets cost $170. One hat, three T-shirts, and two jackets cost $180. Find the prices of the individual items. 

Answer 1

Say you let: x represent hats, y represents shirts, z represent jackets

You have three sets of information/equations and three sets of variables

Set 1: 3x+2y+z=140

Set 2: 2x+2y+2z=170

Set 3: x+3y+2z=180

You want to reduce three equations to two equations by combining any two of the above equations to remove a variable.

In this case, I am taking z out of the picture

Let us do

Set 1 - (Set 2)/2, which gives 2x + y = 55 (Let's call it Combo 1)

&

Set 3 - Set 2, which gives -x + y = 10 (Let's call it Combo 2)

Now, you can combine those two equations in a way to remove either x or y

I'll be taking y out of the picture

So we do

Combo 1 - Combo 2, which gives 3x=45

x=15

Now plug x=15 in to any Combo equation that we made to find y.

Let's use Combo 2 with x = 15

-15 + y = 10

y = 25

Now plug x = 15 and y = 25 in to any Set equation we originally have to find z

Let's use Set 1 with x = 15 and y = 25

3(15) + 2( 25) + z = 140

45 + 50 + z = 140

95 + z = 140

z=45

So hat cost $15, shirt cost $25, and jacket cost $45

Hope that helps!

Answer 2

We set up our system of equations.

h = price of hat, t = price of t shirt, and j = price of jacket

a) 3h + 2t + j = 140

2h + 2t + 2j = 170 %2 b) h + t + j = 85

c) h + 3t + 2j = 180

This is a system of three variables and three equations, so we have to begin reducing the number of variables we have.

Eliminating the j variable...

Equation a) - b) yields d) 2h + t = 55

Equation c) - 2 b) yields e) -h + t = 10

Eliminating the t variable...

Equation d)-e) yields 3h = 45, h=15

Plugging h into d) yields t = 25

Plugging h and t into a) yields j = 45

I hope this helps!