Question

What is the equation of the line that passes through (-2,3) and is parallel to 2x+3y=6?

Answer 1

2x + 3y = 5

Explanation:

`\bold{METHOD\ 1:}`

The slope-intercept form of an equation of a line:

`y=mx+b`

m - slope

b - y-intercept

Let
`k:y=m_1x+b_1,\ l:y=m_2x+b_2`

then

`l\ \parallel\ k\iff m_1=m_2\\\\l\ \perp\ k\iff m_1m_2=-1\to m_2=-(1)/(m_1)`

We have the equation of a line:

`2x+3y=6`

Convert to the slope-intercept form:

`2x+3y=6` subtract 2x from both sides

`3y=-2x+6` divide both sides by 3

`y=-(2)/(3)x+2\to m_1=-(2)/(3)`

therefore the slope is
`m_2=-(2)/(3)`

Put the value of the slope and the coordinates of the point (-2, 3) to the equation of a line:

`3=-(2)/(3)(-2)+b`

`3=(4)/(3)+b` subtract 4/3 from both sides

`(9)/(3)-(4)/(3)=b\to b=(5)/(3)`

Finally:

`y=-(2)/(3)x+(5)/(3)`

Convert to the standard form (Ax + By = C):

`y=-(2)/(3)x+(5)/(3)` multiply both sides by 3

`3y=-2x+5` add 2x to both sides

`2x+3y=5`

`\bold{METHOD\ 2:}`

Let
`k:A_1x+B_1y=C_1,\ l:A_2x+B_2y=C_2`.

Lines k and l are parallel iff

`A_1=A_2\ \wedge\ B_1=B_2\to(A_2)/(A_1)=(B_2)/(B_1)`

We have the equation:

`2x+3y=6\to A_1=2,\ B_1=3`

then the equation of a line parallel to given lines has the equation:

`2x+3y=C`

Put the coordinates of the point (-2, 3) to the equation:

`C=2(-2)+3(3)\\\\C=-4+9\\\\C=5`

Finally:

`2x+3y=5`