1. ## linear equations

students of a class are made to stand in raws.if 4 students are extra in a raw,there would be 2 raws less.if 4 student are less in a raw ,there woulde be 4 more raws,find the number of students?

2. Hello, anjana!

This one was tricky to set-up . . .

Students of a class are made to stand in rows.
(a) If there are 4 more students in each row, there would be 2 rows less.
(b) If there are 4 less students in each row, there would be 4 more rows.
Find the number of students.

Let $\displaystyle X$ = number of students in the class.
Let $\displaystyle n$ = number of students in each row.
Let $\displaystyle r$ = number of rows.

Then we have: .$\displaystyle X \:=\:n\cdot r$ [1]
. . This is the original seating: $\displaystyle n$ in each row and $\displaystyle r$ rows.

Case (a): 4 more students per row, 2 fewer rows.
Then we have: .$\displaystyle X \:=\n + 4)(r - 2)$ [2]

Case (b): 4 less students per row, 4 more rows.
Then we have: .$\displaystyle X\:=\n-4)(r + 4)$ [3]

Equate [1] and [2]: .$\displaystyle nr \:=\n + 4)(r - 2)\quad\Rightarrow\quad 2r - n \:=\:4$

Equate [1] and [3]: .$\displaystyle nr\:=\n - 4)(r + 4)\quad\Rightarrow\quad r - n \:=\:-4$

Solve this system of equations and we get: .$\displaystyle r = 8,\;n = 12$

Therefore: .$\displaystyle X \:=\:12\cdot8\:=\:\boxed{96\text{ students}}$

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Check

Orignally, the 96 students were seated 12 in each row.
. . There were $\displaystyle 8$ rows.

With 4 more in each row, there were 16 in each row.
. . There were $\displaystyle 6$ rows . . . two less rows.

With 4 less in each row, there were 8 in each row.
. . There were $\displaystyle 12$ rows . . . 4 more rows.

We nailed it!