SOLUTION: The pair of equations 3x + 4y = k, 9x +12y = 0 has infinitely many solution if k = ?

Question 924057: The pair of equations 3x + 4y = k, 9x +12y = 0 has infinitely many solution if k = ?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
for there to be an infinnite number of solutions, the lines have to be identical which means they are effectively the same line and occupy the same space on a graph.

this means that, when the lines are in slope intercept form, they have to have the same slope and the same y-intercept.

the equations are:

3x + 4y = k
9x + 12y = 0

in standard form, the lines are identical to each other if they are exact multiples of each other.

when k = 0, these equations will be exact multiples of each other.

you will get:

3x + 4y = 0
9x + 12y = 0

convert each of these eqautions to slope intercept form to confirm they are identical.

3x + 4y = 0 becomes y = (-3/4)x + 0

9x + 12y = 0 becomes y = (-9/12)x + 0 which can be simplified to (-3/4)x + 0

the 0 becomes silent and the equations become:

y1 = (-3/4)x

y2 = (-3/4)x

the slope is the same and the y-intercept is the same so the lines are identical and will both graph as the same line meaning all points on both lines will be common to each other.

your solution is that k = 0

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