Question

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?

Answer 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,, :)

Answer 2

the answer