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School rules permit no fewer than 2 teachers per 25 students. There are at least 245 students enrolled in the school. If [tex]\( x \)[/tex] represents teachers and [tex]\( y \)[/tex] represents students, which system of linear inequalities can be used to determine the possible number of teachers and students at the school?

A. [tex]\( 2y \geq 25x \)[/tex] and [tex]\( y \geq 245 \)[/tex]
B. [tex]\( 2y \leq 25x \)[/tex] and [tex]\( y \geq 245 \)[/tex]
C. [tex]\( 25y \leq 2x \)[/tex] and [tex]\( y \geq 245 \)[/tex]
D. [tex]\( 25y \geq 2x \)[/tex] and [tex]\( y \geq 245 \)[/tex]


Sagot :

To determine the system of linear inequalities that represents the school scenario, let's break down the problem step-by-step:

1. Interpreting the Teacher-Student Ratio Requirement:
School rules mandate no fewer than 2 teachers for every 25 students. This requirement implies that the ratio of teachers to students must be at least [tex]\( \frac{2}{25} \)[/tex].

To express this as an inequality:
- Let [tex]\( x \)[/tex] represent the number of teachers.
- Let [tex]\( y \)[/tex] represent the number of students.
- The school rule can be written in terms of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] as [tex]\( \frac{2}{25} \leq \frac{x}{y} \)[/tex].

To eliminate the fraction, we can cross multiply:
[tex]\[ 2y \geq 25x \][/tex]

2. Interpreting the Minimum Number of Students:
There are at least 245 students enrolled in the school.

This can be straightforwardly written as:
[tex]\[ y \geq 245 \][/tex]

Putting these two pieces of information together, we form the system of linear inequalities:
[tex]\[ 2y \geq 25x \quad \text{and} \quad y \geq 245 \][/tex]

This represents the conditions under which the number of teachers and students at the school must operate.

Therefore, the correct system of linear inequalities is:
[tex]\[ 2y \geq 25x \quad \text{and} \quad y \geq 245 \][/tex]

The correct option is:
[tex]\[ 2 y \geq 25 x \quad \text{and} \quad y \geq 245 \][/tex]