Question: A dog groomer charges $50 to groom a small dog and $75 to groom a large dog. One Saturday, the…

A dog groomer charges $50 to groom a small dog and $75 to groom a large dog. One Saturday, the groomer groomed 9 dogs for a total income of $500. How many small dogs and how many large dogs were groomed?

Solution

A dog groomer charges $50 to groom a small dog and $75 to groom a large dog. One Saturday, the groomer groomed 9 dogs for a total income of $500. We need to determine how many small dogs and large dogs were groomed. Let’s define: - \( x \) as the number of small dogs. - \( y \) as the number of large dogs. We can set up the following system of equations: \[ \begin{cases} x + y = 9 & \text{(total number of dogs)} \\ 50x + 75y = 500 & \text{(total income)} \end{cases} \] First, solve the first equation for \( y \): \[ y = 9 - x \] Substitute \( y = 9 - x \) into the second equation: \[ 50x + 75(9 - x) = 500 \] Simplify and solve for \( x \): \[ 50x + 675 - 75x = 500 \] \[ -25x + 675 = 500 \] \[ -25x = 500 - 675 \] \[ -25x = -175 \] \[ x = 7 \] Using the value of \( x \) in the equation \( y = 9 - x \): \[ y = 9 - 7 \] \[ y = 2 \] Therefore, the groomer groomed 7 small dogs and 2 large dogs.