What type of lines are indicated by these 2 equations?

Question

2x+3y=6 and -4x-6y=-12

(my choices were perpendicular lines, parallel lines, or coinciding lines - yet is there a method to solving and getting the coordinates for the lines? or can you just look at the equations and know?

Answer 1

*Set both equations in the slope-intercept form > y = mx+b

1. 2x+3y = 6
First: subtract 2x from both sides >

2x - 2x + 3y = -2x + 6
3y = -2x + 6

Sec: solve for "y" by isolating it on one side > divide everything by 3 >

3y/3 = (-2/3)x + 6/3
y = (-2/3)x + 2

2. - 4x - 6y = - 12
First: add 4x to both sides >

- 4x + 4x - 6y = 4x - 12
-6y = 4x - 12

Sec: divide both sides by -6 >

-6y/-6 = (4/-6)x - 12/-6
y = (- 4/6)x + 2

*Simplify - 4/6 > y = (- 2/3)x + 2

The lines are coinciding because, they are the same.

Answer 2

Divide the second by (-2) => yields back the first equation; therefore the lines are the same, they coincide.

In general,
Solve each for y. The subsequent equations will be in Slope-Intercept form: y = mx + b.

Lines are parallel if the slopes, m1 & m2, are equal.
Lines are perpendicular if the slopes are negative reciprocals of each other, (m1)(m2) = -1.
Lines coincide if the equations are the same or one is a multiple of the other.

y = (-2/3)x + 2 (i)
y = (-2/3)x + 2 (ii)

Answer 3

By inspection they are both straight lines; so solve each for y in the form of a straight line: y = mx + b
the first one: 3y=-2x +6 or y=-(2/3)x+2
the second one: -6y=4x-12 or y=(-4/6)x-12/(-6) or y = -(2/3)x+2
They are identical or collinear

Answer 4

These are coinciding lines. If you set both equations in slope-intecept form and simplify, you will find that they are the same. You do not have to graph this to find it, just do the algebra.