3. In the same opera theater personal balcony sections with three rows of seating are being mapped as well. In these sections there must be an odd number of seats in each row and each row must have two more seats
than the one before it. The last stipulation is that the front row must have one quarter the total seats in the
back 2 rows combined. How many seats will be in each row?

Number of seats in first row = 2

Number of seats in 2nd row = 4

Number of seats in 3rd row = 6

Step-by-step explanation:

Let's say number of seats in first row is x.

Now we are told that each row must have two more seats

than the one before it.

Thus, the remaining 2 rows have;

(x + 2) and (x + 4) seats respectively.

We are told that that the front row must have one quarter the total seats in the

back 2 rows combined.

Thus;

x = ¼((x + 2) + (x + 4))

Multiply both sides by 4 to give;

4x = x + 2 + x + 4

4x = 2x + 4

4x - 2x = 4

2x = 4

x = 4/2

x = 2

Thus;

Number of seats in first row = x = 2

Number of seats in 2nd row = (x + 2) = 2 + 2 = 4

Number of seats in 3rd row = (x + 4) = 2 + 4 = 6

starting with:    

5 + x - 12 = 2x - 7   

5 + x - 12 - x = 2x - 7 - x (subtract x from both sides)   

5 - 12 = x - 7 (combine and cancel x's from both sides)   

5 - 12 + 7 = x - 7 + 7 (add 7 to both sides)   

0 = x (add numbers together on both sides)       

so in order for the equation to be true, x must be 0.   

trying x = -0.5   

5 + x - 12 = 2x - 7 (original equation)   

5 + -0.5 - 12 = 2(-0.5) - 7 (plug in value of x)   

5 + -0.5 - 12 = -1 - 7 (did multiplication)   

-7.5 = -8 (did addition on both sides)       

since -7.5 is obviously not equal to -8, the value of -0.5 for x is wrong.   

trying x = 0   

5 + x - 12 = 2x - 7 (original equation)   

5 + 0 - 12 = 2(0) - 7 (plug in value of x)   

5 + 0 - 12 = 0 - 7 (did multiplication)   

-7 = -7 (did addition on both sides)       

since -7 equals -7, that demonstrates that the value of 0 for x is correct.   

trying x = 1   

5 + x - 12 = 2x - 7 (original equation)   

5 + 1 - 12 = 2(1) - 7 (plug in value of x)   

5 + 1 - 12 = 2 - 7 (did multiplication)   

-6 = -5 (did addition on both sides)       

since -6 is obviously not equal to -5, the value of 1 for x is wrong.

chemists are often responsible for designing a chemical reaction and analyzing the products obtained from it. one significant part of designing a chemical reaction is determining the amount of each reactant that is needed and the amount of each product that will be produced. in order to do so, chemists use stoichiometry, the quantitative relationships between reactants and products.