We are given first equation: −2x+3y=12.
Converting it in slope-intercept form
3y=2x + 12
y= 2/3 x + 12/3
y= 2/3 x + 4.
Slope of the given equation is 2/3.
Note: When slopes are same, lines would be parallel.
When slope are negative reciprocals, lines would be perpendicular.
When neither same nor negative reciprocal, the lines neither parallel nor perpendicular.
1) First option : -2x+y=12.
In slope-intercept form y = 2x +12.
Slope is 2 there.
2) 3x+2y=-2
In slope-intercept form y = -3/2 x - 1.
Slope = -3/2.
Slope are negative reciprocal to slope of −2x+3y=12 equation.
3) y=2/3x-1
Slope = 2/3.
4) -2x+3y=11
y = 2/3 x + 11/3.
Solution
To solve this problem we will use the form of line y=mx+c with slope of the line= m.
Two lines are parallel if their slopes are equal that is m1=m2
And two lines are perpendicular if the product of their slopes = -1 that is m1.m2= -1
given line -2x+3y=12 ⇒ 3y=12+2x ⇒y=4+2x/3 ⇒ slope(m1) = 2/3
(1) L1: -2x+y=12 ⇒ y=2x+12 ⇒ slope (m2)= 2
By observation m1≠m2 and m1×m2≠ -1
∴ this line is neither parallel nor perpendicular.
(2) L2: 3x+2y= -2 ⇒ y=-3x/2-1 ⇒ slope (m2)= -3/2
By observation m1×m2= -1
∴ this line is perpendicular to given line.
(3) L3: y=23x-1 ⇒ slope (m2)= 23
By observation m1≠m2 and m1×m2≠ -1
∴ this line is neither parallel nor perpendicular
(4) L4: -2x+3y=11 ⇒ y=2x/3+11/3 ⇒ slope (m2)= 2/3
By observation m1=m2
∴ this line is parallel to given line.
