the system of linear equations below has a unique solution for all but one value of a:

6 x−12 y=−6
58 x+a y=−58

What is this exceptional value for a?

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  1. unique solution means the lines intersected at only one point (intersecting lines),, thus lines that do not intersect don not have a solution (parallel lines)
    if two lines are parallel, then their slopes must be equal,, thus get their slopes by writing them into the form:
    y = mx + b
    where m = slope, and b = y-intercept
    *for 6x − 12y = −6,
    6x − 12y = −6
    -12y = -6 - 6x
    y = (1/2)x + (1/2)
    thus m = 1/2
    *for 58x+ ay = −58,
    58x+ ay = −58
    ay = -58 - 58x
    y = (-58/a)x - 58/a
    thus m = -58/a

    equating and solving for a:
    1/2 = -58/a
    a = -116

    so there,, :)

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