Math Question, So Confused

Question

Find the value of d such that the system of equations does not have a solution.

-10x+dy=-1

-8x+12y=-19

Answer 1

If a system of two linear equations does not have a solution, that means if you graph the equations the lines will be parallel with difference y-intercepts. If they had the same y-intercept the lines would really be the same line.

So put each equation in slope intercept form, find the value of "d" that gives the equations the same slope, and make sure they have different y-intercepts.

For the first equation:

dy = 10x-1

y = (10/d)x -(1/d).

m₁= 10/d

For equation two:

12y = 8x-19

y = (8/12)x - (19/12)

m₂=2/3

For the slopes to be equal:

10/d = 2/3

d=15

With a value of d = 15 the y-intercept of the 1st equation is (0,-1/15) which is different than (0,-19/12). So a value of d=15 ensures that the system will not have a solution (because the lines are parallel with different y-intercepts).

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