Question

1. government the texas state legislature is comprised of state senators and state representatives. there is a greater number of representatives than senators. the sum of the number of representatives and the number of senators is 181. the difference of the number of representatives and number of senators is 119.

a. write a system of equations to find the number of state representatives, ( r ), and senators, ( s ).
b. how many senators and how many representatives make up the texas state legislature?

Explanation:

Part a

Step1: Define variables and sum equation

Let \( r \) be representatives, \( s \) be senators. Sum: \( r + s = 181 \)

Step2: Define difference equation

Difference: \( r - s = 119 \)

Step1: Add the two equations

Add \( r + s = 181 \) and \( r - s = 119 \):
\((r + s) + (r - s) = 181 + 119\)
Simplify: \( 2r = 300 \)

Step2: Solve for \( r \)

Divide by 2: \( r = \frac{300}{2} = 150 \)

Step3: Substitute \( r \) to find \( s \)

Use \( r + s = 181 \): \( 150 + s = 181 \)
Subtract 150: \( s = 181 - 150 = 31 \)

Answer:

The system is \(

$$\begin{cases} r + s = 181 \\ r - s = 119 \end{cases}$$

\)

Part b