# If ax + 6y = 4 , 3x + by = 1 and 5x + 3y = 8 are parallel lines, what are a and b equal to?

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justaguide | College Teacher | (Level 2) Distinguished Educator

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Parallel lines have the same slope. We have to write the given lines in the form y = mx + c, where m is the slope and c is the y-intercept.

For ax + 6y = 4

=> 6y = -ax + 4

=> y = (-a / 6) x + 4

For 3x + by =1

=> by = -3x + 1

=> y = (-3/b) x + 1/b

For 5x + 3y = 8

=> 3y = 8 – 5x

=> y = (-5/3) x + (8/5)

Now as the slope of all the three lines has to be the same -5/3 = -a/6 = -3/b

-5/3 = -a/6

=> a = 5*6/3 = 10

-5/3 = -3/b

=> b = 3*3/5

=> b = 9/5

Therefore a = 10 and b = 9/5.

msrenfroe | High School Teacher | eNotes Newbie

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To convert easily from the Ax + By = C formula to y = mx + b.  Someone came up with the saying "Negative Abe crossed the crazy bridge."  (-A/B)x + (C/B)  This takes us easily from general form to slope-intercept.  And you can put it into the "y=" of a TI-83 calculator.

The hard part of answering this question is to commit to memory that parallel lines have the same slope.

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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If the lines that have the equations ax + 6y = 4 , 3x + by = 1 are parallel, their slopes have to be equal.

we'll put the equations in the slope inntercept form:

6y = -ax + 4

We'll divide by 6:

y =  -ax/6 + 2/3

The 1st equation has changed into

y =  -ax/6 + 2/3, where m1 = -a/6

The second equation is:

by = -3x + 1

y = -3x/b + 1/b

m2 = -3/b

We'll put m1 = m2:

-a/6 = -3/b

We'll cross multiply and we'll get:

ab  =18

The 3rd equation will be:

3y = -5x + 8

y = -5x/3 + 8/3

m3 = -5/3

m3 = m1

-5/3 = -a/6

5/1 = a/2

a = 10 => b = 18/10

b = 1.8

The coefficients a and b are: a = 10 ; b = 1.8.

neela | High School Teacher | (Level 3) Valedictorian

Posted on

If ax + 6y = 4 , 3x + by = 1 and 5x + 3y = 8 are parallel lines, what are a and b equal to?

We write both lines in the  slope intercept form like y = mx+c, where m is the slope and c  is the the y intercept of the line y = mx+c.

ax+6y = 4.

We subtract ax from both sides.

6y = 4-ax.

We divide both sides by 6

y = (4-ax)/6  =  (-a/6)x +4/6.

So y = -(a/6)x+2/3.....(1).

No consider the line 3x+by = 1.

3x+by - 3ax = 1-3x

by = (1-3x).

by/b = (1-3x)/b = -(3/b)x+1/b

y = -(3/b)x+1/b..........(2).

Now we consider the line 5x+3y = 8

=> 3y = 8-5x

=> y = (8-5x)/3 = -(5/3)x+8.

=> y = -(5/3)x+8.........(3)

Since the (1) and (2) and (3) are parallel, their slopes -(a/6) ,  -(3/b)  and (-5/3) must be equal.

=> -a/6 = -3/b = -5/3.

=> a/6 = 5/3 = 10 and 3/b = 5/3

So a = 5*6/2 = 10 and b = 3*3/5  = 1.8

So a = 10 and b = 1.8.

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