9 small boxes and 2 large boxes are 110. 10 small boxes and 14 large boxes are 346. what is the cost of 1 small box and 1 large box

Question

Trying to help my autistic teen with her math  been way too long for me and im lost.  Getting confused halfway through problem 

Answer 1

Set up a system of equations:

 

9s + 2l = $110

10s + 14l = $346

 

You can solve by elimination or substitution. Most students find substitution easier, so we'll do that.

 

First, rearrange either of the two equation to get a single variable by itself.

 

9s + 2l = $110

⇒ 9s = $110 - 2l

⇒ s = ($110 - 2l) / 9

 

Then substitute into the other equation and solve for l:

10s + 14l = $346

⇒ 10(($110 - 2l) / 9) + 14l = $346

⇒ ($1100 - 20l) / 9) + 14l = $346

⇒ $1100 - 20l + 126l = $3114

⇒ 106l = $2014

⇒ l = $19

 

Then substitute into either of the two original equations to solve for the other variable:

9s + 2l = $110

⇒ 9s + 2($19) = $110

⇒ 9s + $38 = $110

⇒ 9s = $72

⇒ s = $8

 

What is the cost of 1 small box and 1 large box?

1($8) + 1($19) = $27

Answer 2

There are two relationships described, let's write equations to match them: 

 

9S + 2L = 110

10S + 14L = 346 

 

Now let's solve one of these for ONE of the variables. (I'm going to chose the first one because it has smaller numbers)

1.) move the 9s to the right side by subtracting it from both sides: 

2L = 110 - 9S 

2.) Divide both sides of the equation by 2:

L = (110 -9S)

            2

 

now let's substitute this value (or expression) for L into the other equation: 

 

10S + 14 (110 - 9S)  = 346

                     2

Now we can solve for S by first multiplying everything by 2 to get rid of the fraction and distributing the 14 through the parentheses:

 

20S + 1540 - 126 S = 692 

 

Now get like terms together, S's on the left and constants on the right: 

 

-106S = -848

 

Now divide both sides by -106 

 

s = 8

 

now we can substitute this solution into one of the original equations: 

 

9(8) +2L = 110

72 + 2L = 110

2L = 38

L = 19

 

Now let's check to make sure our answer makes sense: 

 

9(8) + 2(19) = 110 ???? does it? YES

10(8) +14(19) = 346 ??? yes!