Using system of equations to break even

A fashion designer makes and sells hats.  The material for each hat costs $5.50.  The hats sell for $12.50 each.  The designer spends $1400 on advertising.  How many hats must the designer sell to break even?

x = the number of hats sold
y = the number of dollars of expense or income

y = 12.5x   {the income is $12.50 times the hats sold}
y = 5.5x + 1400   {the expense is $5.50 times the hats sold plus $1400 in advertising}

To break even means for the income to be equal to the expense.

12.5x = 5.5x + 1400   {substituted 12.5x, in for y, into bottom equation, setting income equal to expense}
7x = 1400   {subtracted 5.5x from each side}
x = 200   {divided each side by 7}

the designer must sell 200 hats to break even

- Algebra House