Algebra question

Question

a theater artist charges 150 per show for a 4 hour show and 85 for a 2 hour show. last week the artist performed 5 less 4 hour shows than 2 hour shows. if the artists income was 1835, the total number of shows performed by the artist is?

Answer 1

Word problems are tricky! Step one should be defining the variables in the question. Here we see there are two variables - 4 hour shows and 2 hour shows. Let's call the 4 hour shows "x" and the 2 hour shows "y".

We see that "last week the artist performed 5 less 4 hour shows than 2 hour shows". As an equation with variables, we see that y - x = 5.

The second equation in the word problem is that of the artist's rates. The total income (we'll call it "P") in the last week was 1835. We know the prices she charges for each show as well. For a 4 hour show, the artist charges 150. And for a 2 hour show, she charges 85. Therefore, the equation can be expressed as:

P = 150x + 85y

1835 = 150x + 85y /substituting the income given for the last week.

We now have a classic system of equations problem with the two equations y - x = 5 and 1835 = 150x + 85y. This can be solved as follows:

y - x = 5

y = 5 + x /solving for y to substitute into the other equation

1835 = 150x + 85(5 + x) /substituting the first equation solved for "y" into the second equation

1835 = 150x + 425 + 85x /distributing the parenthetical expression

1410 = 150x + 85x /subtracting 425 from both sides

1410 = 235x /combining like terms

6 = x /solving for x by dividing both sides by 235

We then plug 6 = x into the first equation.

y - 6 = 5

y = 11 /adding 6 to both sides

We know that y = 11 and 6 = x but remember to always look back at the question given in the problem. What is the total number of shows performed by the artist? Well, 6 + 11 = 17.

17 shows!