Question

Are the lines `3x-2y=-2` and `6x-4y=0` parallel, perpendicular, or neither?

Answer 1

lines are parallel.

Explanation:

The equation of a line in `color(blue)"slope-intercept form"` is

`color(red)(|bar(ul(color(white)(a/a)color(black)(y=mx+b)color(white)(a/a)|)))`
where m represents the slope and b, the y-intercept.

The advantage to this form is that m and b may be extracted 'easily'

Consider the 2 lines with slope `m_1" and " m_2`

• If lines are parallel , then `color(red)(|bar(ul(color(white)(a/a)color(black)(m_1=m_2)color(white)(a/a)|)))`

• If perpendicular , then `color(red)(|bar(ul(color(white)(a/a)color(black)(m_1xxm_2=-1)color(white)(a/a)|)))`

Rearrange the 2 equations into `color(blue)"slope-intercept form"` and 'extract' their slopes.

`color(red)"3x-2y=-2"`

Subtract 3x from both sides

hence `cancel(3x)-2y-cancel(3x)=-3x-2rArr-2y=-3x-2`

divide both sides by -2

`(cancel(-2) y)/cancel(-2)=(-3)/(-2)x-2/-2rArry=3/2x+1`

`rArrcolor(magenta)" this line has slope =3/2"`

`color(red)"6x-4y=0"`

Subtract 6x from both sides

`cancel(6x)-4y-cancel(6x)=0-6xrArr-4y=-6x`

divide both sides by -4

`(cancel(-4) y)/cancel(-4)=(-6)/(-4)xrArry=3/2x`

`rArrcolor(magenta)"this line has slope =3/2"`

Since the slope of both lines are equal then the 2 lines are parallel.