Question

For what value of k are the two lines 2x +ky =3 and x+y=1
(a)parallel?
(b)perpendicular?

Answer 1

if two lines are parallel, that means their "slope" is?

Answer 2

the same

Answer 3

right

so.... let us put those two fellows in slope-intercept form

so \(\bf \begin{cases}
2x+ky=3\to ky=-2x+3\to y=\cfrac{-2x+3}{k}\to &y=-\cfrac{2}{k}+\cfrac{3}{k}
\\ \quad \\
x+y=1\to y=-x+1\to &y=-1x+1
\end{cases}\)


notice the slopes of each of them

Answer 4

yeah the slop of the first one is -2/k and the second one is -1

Answer 5

so
the 1st one has a slope of -2/k
the 2nd one, has a slope of -1


what should "k" need to be, for them to be equal? well \(\bf -\cfrac{2}{k}=-1\implies -\cfrac{2}{-1}=k\implies 2=k\)

meaning, that, if "k" is 2, then, the EQUATion is true, thus the slopes are equal
and thus both lines are parallel, when "k" is 2

Answer 6

now... for two lines to be perpendicular
their slopes need to be the NEGATIVE RECIPROCAL, that is \(slope=\cfrac{a}{{\color{blue}{ b}}}\qquad negative\implies -\cfrac{a}{{\color{blue}{ b}}}\qquad reciprocal\implies - \cfrac{{\color{blue}{ b}}}{a}\)

Answer 7

and we can negativize and reciprocalize either for that
and then find "k"

Answer 8

so i do the same thing but just have the slopes as a reciprocal

Answer 9

right.. .one sec

Answer 10

\(\bf -\cfrac{1}{{\color{blue}{ 1}}}\qquad negative\implies \cfrac{1}{{\color{blue}{ 1}}}\qquad reciprocal\implies \cfrac{{\color{blue}{ 1}}}{1}\implies 1
\\ \quad \\
\textit{thus if we equate the 1st one, with THAT negative reciprocal}
\\ \quad \\
-\cfrac{2}{k}=1\implies -\cfrac{2}{1}=k\implies 2=k\)

Answer 11

oh ok thnx Cx i get it now

Answer 12

hmmm actually, should be -2, shoot, missing the negative there

\(\bf -\cfrac{1}{{\color{blue}{ 1}}}\qquad negative\implies \cfrac{1}{{\color{blue}{ 1}}}\qquad reciprocal\implies \cfrac{{\color{blue}{ 1}}}{1}\implies 1
\\ \quad \\
\textit{thus if we equate the 1st one, with THAT negative reciprocal}
\\ \quad \\
-\cfrac{2}{k}=1\implies -\cfrac{2}{1}=k\implies -2=k\)

Answer 13

so i just had to do the reciprocal? i thought you had to change the signs too

Answer 14

because the -2/k was already negative so wouldnt it be k/2 =1 @jdoe0001

Answer 15

nevermind