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Math Help - linear equations

  1. #1
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    Question 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?
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  2. #2
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    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 X = number of students in the class.
    Let n = number of students in each row.
    Let r = number of rows.

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

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

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


    Equate [1] and [2]: . n + 4)(r - 2)\quad\Rightarrow\quad 2r - n \:=\:4" alt="nr \:=\n + 4)(r - 2)\quad\Rightarrow\quad 2r - n \:=\:4" />

    Equate [1] and [3]: . n - 4)(r + 4)\quad\Rightarrow\quad r - n \:=\:-4" alt="nr\:=\n - 4)(r + 4)\quad\Rightarrow\quad r - n \:=\:-4" />


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

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

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    Check

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

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

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

    We nailed it!

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