Question

use the information to answer the questions about systems of equations.
eight dogs and 15 cats eat 15.5 pounds of food per day. twenty dogs and 12 cats eat 26 pounds of food per day.
let ( x=) pounds of food eaten by each dog
( y=) pounds of food eaten by each cat

1. write a system of equations to represent this situation. (2 points)
2. solve the system using either substitution or elimination. write your answer below and circle which method you used. show all work. (2 points)

solution: substitution elimination

Explanation:

Step1: Write the system of equations

For the first situation: \(8x + 15y=15.5\) (since 8 dogs eat \(8x\) pounds and 15 cats eat \(15y\) pounds, total 15.5 pounds).
For the second situation: \(20x + 12y = 26\) (since 20 dogs eat \(20x\) pounds and 12 cats eat \(12y\) pounds, total 26 pounds).

Step2: Use the elimination method

Multiply the first equation \(8x + 15y=15.5\) by 5: \(40x+75y = 77.5\).
Multiply the second equation \(20x + 12y = 26\) by 2: \(40x+24y=52\).
Subtract the second - new equation from the first - new equation:
\((40x + 75y)-(40x + 24y)=77.5 - 52\).
\(40x+75y - 40x - 24y=25.5\).
\(51y=25.5\).
Solve for \(y\): \(y=\frac{25.5}{51}=0.5\).

Step3: Substitute \(y = 0.5\) into one of the original equations

Substitute \(y = 0.5\) into \(8x + 15y=15.5\).
\(8x+15\times0.5 = 15.5\).
\(8x+7.5=15.5\).
\(8x=15.5 - 7.5\).
\(8x=8\).
Solve for \(x\): \(x = 1\).

Answer:

1. The system of equations is \(

$$\begin{cases}8x + 15y=15.5\\20x + 12y = 26\end{cases}$$

\).

1. Using the elimination method, \(x = 1\) (pounds of food eaten by each dog) and \(y=0.5\) (pounds of food eaten by each cat).