1. Find the equation of the line passing through the point `A(5,4)` and parallel to the line `2x+3y+7=0`

Solution:
Here Point `(x_1,y_1)=(5,4)` and line `2x+3y+7=0` (given)

When two lines are parallel, their slopes are equal.

We shall first find the slope of line `2x+3y+7=0`

`2x+3y+7=0`

`:. 3y=-2x-7`

`:. y=-(2x)/(3)-7/3`

`:.` Slope `=-2/3` and hence slope of the line parallel to this line is also `-2/3`



Here Point `(x_1,y_1)=(5,4)` and Slope `m=-2/3` (given)

`:. y-4=-2/3(x-5)`

`:. 3(y-4)=-2(x-5)`

`:. 3y -12=-2x +10`

`:. 2x+3y-22=0`

Hence, The equation of line is `2x+3y-22=0`

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2. Find the equation of the line passing through the point `A(1,1)` and parallel to the line `2x-3y+2=0`

Solution:
Here Point `(x_1,y_1)=(1,1)` and line `2x-3y+2=0` (given)

When two lines are parallel, their slopes are equal.

We shall first find the slope of line `2x-3y+2=0`

`2x-3y+2=0`

`:. 3y=2x+2`

`:. y=(2x)/(3)+2/3`

`:.` Slope `=2/3` and hence slope of the line parallel to this line is also `2/3`



Here Point `(x_1,y_1)=(1,1)` and Slope `m=2/3` (given)

`:. y-1=2/3(x-1)`

`:. 3(y-1)=2(x-1)`

`:. 3y -3=2x -2`

`:. 2x-3y+1=0`

Hence, The equation of line is `2x-3y+1=0`

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3. Find the equation of the line passing through the point `A(2,3)` and parallel to the line `2x-3y+8=0`

Solution:
Here Point `(x_1,y_1)=(2,3)` and line `2x-3y+8=0` (given)

When two lines are parallel, their slopes are equal.

We shall first find the slope of line `2x-3y+8=0`

`2x-3y+8=0`

`:. 3y=2x+8`

`:. y=(2x)/(3)+8/3`

`:.` Slope `=2/3` and hence slope of the line parallel to this line is also `2/3`



Here Point `(x_1,y_1)=(2,3)` and Slope `m=2/3` (given)

`:. y-3=2/3(x-2)`

`:. 3(y-3)=2(x-2)`

`:. 3y -9=2x -4`

`:. 2x-3y+5=0`

Hence, The equation of line is `2x-3y+5=0`

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4. Find the equation of the line passing through the point `A(2,-5)` and parallel to the line `2x-3y-7=0`

Solution:
Here Point `(x_1,y_1)=(2,-5)` and line `2x-3y-7=0` (given)

When two lines are parallel, their slopes are equal.

We shall first find the slope of line `2x-3y-7=0`

`2x-3y-7=0`

`:. 3y=2x-7`

`:. y=(2x)/(3)-7/3`

`:.` Slope `=2/3` and hence slope of the line parallel to this line is also `2/3`



Here Point `(x_1,y_1)=(2,-5)` and Slope `m=2/3` (given)

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

`:. 3(y+5)=2(x-2)`

`:. 3y +15=2x -4`

`:. 2x-3y-19=0`

Hence, The equation of line is `2x-3y-19=0`

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