1. Find the equation of the line passing through the point `A(1,3)` and parallel to line passing through the points `B(3,-5)` and `C(-6,1)`

Solution:
Given points are `A(1,3)`, `B(3,-5)` and `C(-6,1)`

When two lines are parallel, their slopes are equal.

We shall first find the slope of `B(3,-5)` and `C(-6,1)`

Points are `B(3,-5),C(-6,1)`

`:. x_1=3,y_1=-5,x_2=-6,y_2=1`

Slope of the line, `m=(y_2-y_1)/(x_2-x_1)`

`:. m=(1+5)/(-6-3)`

`:. m=(6)/(-9)`

`:. m=-2/3`

`:.` Slope `=-2/3`

Hence, slope of the line parallel to this line is also `-2/3` `(:' m_1=m_2)`



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

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

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

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

`:. 2x+3y-11=0`

Hence, The equation of line is `2x+3y-11=0`

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2. Find the equation of the line passing through the point `A(4,-5)` and parallel to line passing through the points `B(3,7)` and `C(-2,4)`

Solution:
Given points are `A(4,-5)`, `B(3,7)` and `C(-2,4)`

When two lines are parallel, their slopes are equal.

We shall first find the slope of `B(3,7)` and `C(-2,4)`

Points are `B(3,7),C(-2,4)`

`:. x_1=3,y_1=7,x_2=-2,y_2=4`

Slope of the line, `m=(y_2-y_1)/(x_2-x_1)`

`:. m=(4-7)/(-2-3)`

`:. m=(-3)/(-5)`

`:. m=3/5`

`:.` Slope `=3/5`

Hence, slope of the line parallel to this line is also `3/5` `(:' m_1=m_2)`



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

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

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

`:. 5y +25=3x -12`

`:. 3x-5y-37=0`

Hence, The equation of line is `3x-5y-37=0`

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3. Find the equation of the line passing through the point `A(-1,3)` and parallel to line passing through the points `B(0,2)` and `C(4,5)`

Solution:
Given points are `A(-1,3)`, `B(0,2)` and `C(4,5)`

When two lines are parallel, their slopes are equal.

We shall first find the slope of `B(0,2)` and `C(4,5)`

Points are `B(0,2),C(4,5)`

`:. x_1=0,y_1=2,x_2=4,y_2=5`

Slope of the line, `m=(y_2-y_1)/(x_2-x_1)`

`:. m=(5-2)/(4-0)`

`:. m=(3)/(4)`

`:.` Slope `=3/4`

Hence, slope of the line parallel to this line is also `3/4` `(:' m_1=m_2)`



Here Point `(x_1,y_1)=(-1,3)` and Slope `m=3/4` (given)

`:. y-3=3/4(x+1)`

`:. 4(y-3)=3(x+1)`

`:. 4y -12=3x +3`

`:. 3x-4y+15=0`

Hence, The equation of line is `3x-4y+15=0`

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4. Find the equation of the line passing through the point `A(2,-3)` and parallel to line passing through the points `B(1,2)` and `C(-1,5)`

Solution:
Given points are `A(2,-3)`, `B(1,2)` and `C(-1,5)`

When two lines are parallel, their slopes are equal.

We shall first find the slope of `B(1,2)` and `C(-1,5)`

Points are `B(1,2),C(-1,5)`

`:. x_1=1,y_1=2,x_2=-1,y_2=5`

Slope of the line, `m=(y_2-y_1)/(x_2-x_1)`

`:. m=(5-2)/(-1-1)`

`:. m=(3)/(-2)`

`:. m=-3/2`

`:.` Slope `=-3/2`

Hence, slope of the line parallel to this line is also `-3/2` `(:' m_1=m_2)`



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

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

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

`:. 2y +6=-3x +6`

`:. 3x+2y=0`

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

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5. Find the equation of the line passing through the point `A(4,2)` and parallel to line passing through the points `B(1,-1)` and `C(3,2)`

Solution:
Given points are `A(4,2)`, `B(1,-1)` and `C(3,2)`

When two lines are parallel, their slopes are equal.

We shall first find the slope of `B(1,-1)` and `C(3,2)`

Points are `B(1,-1),C(3,2)`

`:. x_1=1,y_1=-1,x_2=3,y_2=2`

Slope of the line, `m=(y_2-y_1)/(x_2-x_1)`

`:. m=(2+1)/(3-1)`

`:. m=(3)/(2)`

`:.` Slope `=3/2`

Hence, slope of the line parallel to this line is also `3/2` `(:' m_1=m_2)`



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

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

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

`:. 2y -4=3x -12`

`:. 3x-2y-8=0`

Hence, The equation of line is `3x-2y-8=0`

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6. Find the equation of the line passing through the point `A(5,5)` and parallel to line passing through the points `B(1,-2)` and `C(-5,2)`

Solution:
Given points are `A(5,5)`, `B(1,-2)` and `C(-5,2)`

When two lines are parallel, their slopes are equal.

We shall first find the slope of `B(1,-2)` and `C(-5,2)`

Points are `B(1,-2),C(-5,2)`

`:. x_1=1,y_1=-2,x_2=-5,y_2=2`

Slope of the line, `m=(y_2-y_1)/(x_2-x_1)`

`:. m=(2+2)/(-5-1)`

`:. m=(4)/(-6)`

`:. m=-2/3`

`:.` Slope `=-2/3`

Hence, slope of the line parallel to this line is also `-2/3` `(:' m_1=m_2)`



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

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

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

`:. 3y -15=-2x +10`

`:. 2x+3y-25=0`

Hence, The equation of line is `2x+3y-25=0`

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