For what value of ‘k’ the pair of equations x-ky=2 and 3x+2y=-50 has a unique solution?

Question

For what value of ‘k’ the pair of equations x-ky=2 and 3x+2y=-50 has a unique solution?

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Serenity 5 months 2021-12-13T02:53:56+00:00 2 Answers 0 views 0

Answers ( )

    0
    2021-12-13T02:55:01+00:00

    Answer:

    Find the value of k for which the system of linear equation

    x +2y = 3 and 5x + k y = -7 has unique solution

    0
    2021-12-13T02:55:11+00:00

    Answer:

    k ≠ -2/3

    Note:

    ★ A linear equation is two variables represent a straight line .

    ★ The word consistent is used for the system of equations which consists any solution .

    ★ The word inconsistent is used for the system of equations which doesn’t consists any solution .

    ★ Solution of a system of equations : It refers to the possibile values of the variable which satisfy all the equations in the given system .

    ★ A pair of linear equations are said to be consistent if their graph ( Straight line ) either intersect or coincide each other .

    ★ A pair of linear equations are said to be inconsistent if their graph ( Straight line ) are parallel .

    ★ If we consider equations of two straight line

    ax + by + c = 0 and a’x + b’y + c’ = 0 , then ;

    • The lines are intersecting if a/a’ ≠ b/b’ .

    → In this case , unique solution is found .

    • The lines are coincident if a/a’ = b/b’ = c/c’ .

    → In this case , infinitely many solutions are found .

    • The lines are parallel if a/a’ = b/b’ ≠ c/c’ .

    → In this case , no solution is found .

    Solution:

    Here,

    The given equations are ;

    x – ky = 2

    3x + 2y = -50

    The given equations can be rewritten as ;

    x – ky – 2 = 0

    3x + 2y + 50 = 0

    Clearly ,

    a = 1 , b = -k , c = -2

    a’ = 3 , b = 2 , c = 50

    Now,

    a/a’ = 1/3

    b/b’ = -k/2

    c/c’ = -2/50 = -1/25

    Now,

    We know that , for unique solution ;

    => a/a’ ≠ b/b’

    => 1/3 ≠ -k/2

    => -k/2 ≠ 1/3

    => k ≠ -2/3

    Hence ,

    k ≠ -2/3 ( k can be any real number other than -2/3 ) .

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