Lines relation

Question

what is the relation between the lines described by the equations -20x+12y=36 and -35x+21y=62? find a third equation in the form ax+by=90 that fits this pattern

Answer 1

First, see if there's a GCF (greatest common factor) for each equation.

 

-20x + 12y = 36; the GCF is 4; divide each term by the GCF

 

-20/4x + 12/4y = 36/4 => -5x + 3y = 9

 

Now, do the same for the second equation:

 

-35x + 21y = 62; the GCF is 7; divide each term by 7:

 

-35/7x + 21/7y = 62/7 => -5x +3y = 8.86 (I'm wondering here if the second equation should be -35x + 21y = 63 rather than 62)

 

If you look at the two equations, you'll note that the coefficients for both x and y are the same.  As a result, the slope of both lines is identical.  If the second equation has 63 as the c term, then the lines are the same.  If not, then the y-intercept of both is different.

 

The third equation must have the same a and b coefficients as the other two (-5, 3) after dividing by its GCF.

 

So:

 

for ax + by = 90, the GCF can be any integer that is a factor of 90;  Let's use 10.

 

Then, we need to find a such that a/10 = -5 and b such that b/10 = 3.

 

a/10 = -5 => a = -50

 

b/10 = 3 => b = 30

 

So one other equation is -50x + 30y = 90

 

Please note that if we had chosen a different factor of 90 (say 3), we would have gotten a different equation that would also be correct.

 

e.g., 

 

ax + by = 90

 

a/3 = -5 => a = -15

 

b/3 = 3 => b = 9

 

-15x + 9y = 90