Question

(1) Find the y-intercept of the line with the slope -5/3 that passes through the point of intersection of the lines: 2014x + 2015y=2016
2019x + 2018y=2017
(2) A class has x boys and y girls. In an exam, the average score for the boys is 63 and the average score for the girls is 70. The average for the whole class is 68. What are the possible values of x and y?

Please give the answers as soon as possible.

Answer 1

(a) first of all we have to find out intersection point of the lines :
$2014x + 2015y = 2016\\2019x+2018y=2017$

add both the given equations ,
(2014 + 2019)x + (2015 + 2018)y = 2016 + 2017
4033x + 4033y = 4033
x + y = 1 , y = (1 - x) -----(1)

put equation (1) in 2014x + 2015y = 2016
2014x + 2015 - 2015x = 2016
-x = 1 => x = -1 so, y = 2

hence, intersection point of given lines is (-1,2)
so, equation of line is given by $\bf{y-y_1=m(x-x_1)}$
here m = slope of line = -5/3
and (x1, y1) = (-1,2)
so, (y - 2) = -5/3(x + 1)
=> 3(y - 2) + 5(x + 1) = 0
=> 3y - 6 + 5x + 5 = 0
=> 5x + 3y - 1 = 0
=> 3y = 1 - 5x
=> y = 1/5 + (-5/3)x
hence, y - intercept of line is 1/5


(b) number of boys = x
number of girls = y
average score by boys = 63
so, total scores by boys = 63x
average scroe by girls = 70
so, total scores by girls = 70y

now, average score of class = 68
(63x + 70y)/(x + y) = 68
=> 63x + 70y = 68x + 68y
=> 2y = 5x , y/x = 5/2
so, if 700 students are in class
then, 500 students are girls and 200 students are boys .