Homework Help

What is the relationship between ax+by = c and bx -ay = c if a,b,c are not zero?

user profile pic

alinnne | Student, Undergraduate | eNoter

Posted January 19, 2011 at 11:13 AM via web

dislike 0 like

What is the relationship between ax+by = c and bx -ay = c if a,b,c are not zero?

3 Answers | Add Yours

user profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted January 19, 2011 at 11:19 AM (Answer #1)

dislike 0 like

Given the equations:

ax + by = c

bx - ay = c

We notice that the above are equations of two lines.

We need to find the relation between the lines (i.e perpendicular, parallel, or neither).

To determine the relations, we will rewrite the equations into the slope form " y= mx + a) where m is the slope.

==> ax + by = c

==> by = -ax + c

==> y (-a/b) x + c/b...............(1)

Then slope for equation (1) is m1 = -a/b.

Now we will rewrite the second equation.

==> bx -ay = c

==> -ay = -bx + c

==> y= (b/a)x - c/a..............(2)

The slope for equation (2) is m2= b/a

Now we notice that m1 and m2 are NOT equal, then they are not parallel.

However, m1*m2 = -a/b * b/a = -1

Then, the relationship between the lines is that they are perpendicular.

user profile pic

kjcdb8er | Teacher | (Level 1) Associate Educator

Posted January 19, 2011 at 11:24 AM (Answer #2)

dislike 0 like

You can examine these equations more closely by putting them in standard form: y = mx + b

 

ax + by = c  -->  y = -a/bx + c/b

bx - ay = c   --> y = b/a x - c/a

So the slope of the first equation is m1 = -a/b, and the slope of the second equation is m2 = b/a = -1/m1

When the slopes of two lines are the negative inverse of one another, they are perpendicular lines.

Therefore, the relationship between these two lines is that they are perpendicular.

user profile pic

neela | High School Teacher | Valedictorian

Posted January 19, 2011 at 11:31 AM (Answer #4)

dislike 0 like

We know that  the equations of the line are

ax+by = c...(1).

bx-ay = c....(2).

We rewrite both equations in the  slope intercept form y = mx+k for, where m is the slope of the line and k = the y intercept.

The line (1): ax+by = c.

=> by = -ax+c

=> by/b = (-a/b)x+(c/b).

y = (-a/b)x+(a/b).........(i)

The line (2): bx-ay = c.

-ay = -ax+c.

-ay/-a = (-b/-a)x+(c/-a).

=> y = (b/a)x -c/a..........(ii)

So from (1) and (2)the slopes of the lines  (1) and  (2) are -a/b and  b/a. The product of the slopes is -1. This indicates that  the lines are perpendicular.

The lines ax-by = c and bx-ay = c are perpendicular.

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes