Question 1039503: School rules permit no fewer than 2 teachers per 25 students. There are at least 245 students enrolled in the school. If x represent teachers and y represents students, which system of linear inequalities can be used to determine the possible number of teachers and students at the school?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if x is the number of teachers and y is the number of students, the rule is that x >= 2/25 * y.
since there are no fewer than 245 students enrolled in the school, then the rule is that y >= 245.
your requirements are:
x >= 2/25 * y
y >= 245.
you can pick a y greater than or equal to 245 and solve for x and then round up to the next nearest integer.
for example, when y = 245, x has to be greater than or equal to 2/25 * y which makes x greater than or equal to 19.6 which you would round up to 20.
suppose you want hire a maximum of 35 teachers?
you want to know how many students that would support.
you use x >= 2/25 * y, where x is equal to 35 and y is what you want to find.
the formula becomes 35 >= 2/25 * y
multiply both sides of the equation by 25/2 to get 35 * 25/2 >= 2/25 * 25/2 * y, which simplifies to 437.5 >= y.
this is the same as y <= 437.5.
this means that y has to be less than or equal to 437.5 which you would round down to 437.
to see if you did this correctly, you would estimate the number of teachers required to support 437 students.
use x >= 2/25 * 437 which becomes x >= 34.96 which you would round up to 35.
your system of equation is:
x >= 2/25 * y
y >= 245
you can graph this system of equations by solving for y in both equations and then graphing the equality portion of the equation and then finding the area on the graph that satisfies the inequality.
x >= 2/25 * y becomes y <= 25/2 * x.
y >= 245 stays the same.
you graph y = 25/2 * x and y = 245 and then look for the area on the graph that satisfies y <= 25/2 * x and y >= 245.
the graph would look like this.
you can see from the graph that when y = 245, x = 19.6.
you would round this up to 20 because x has to be greater than or equal to 19.6 and 20 is the nearest integer that satisfies that inequality.
you can see from the graph that when x = 35, y = 437.5.
you would round this down to 437 because y has to be smaller than or equal to 25/2 * x and 437 is the nearest integer that satisfies that inequality.
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