Find the equation of the lines through the point of intersection of the lines x-y+1=0 and 2x-3y+5=0 and whose distance from the point (3,2) is 7/5

You need to find the coordinates of the point P to write the equation of the line passing through this point.

The point P represents the points of intersection of the lines x-y+1=0 and  2x-3y+5=0, hence you need to find the solution of the system of these two equations.

Multiply the first equation by 2, then subtract then subtract the second equation from this new equation.

2x-2y-2-2x+3y-5=0

Eliminating the variable x yields:

y - 7 = 0 => y = 7

Plugging y = 7 in the first equation yields:

x - 7+ 1 = 0 => x = 6

The coordinates of the point P are (6;7).

The line passing through P is 7 = 6m + b.

The point (3,2) is at the distance d =   from the line 7 = 6m + b , hence you may replace these values in the following formula:

 

The equation of the line passing through P and found at a distance of  from the point (3;2) is 6m + b - 7 = 0, m and b being in the following relation 

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You need to find the coordinates of the point P to write the equation of the line passing through this point.

The point P represents the points of intersection of the lines x-y+1=0 and  2x-3y+5=0, hence you need to find the solution of the system of these two equations.

Multiply the first equation by 2, then subtract then subtract the second equation from this new equation.

2x-2y-2-2x+3y-5=0

Eliminating the variable x yields:

y - 7 = 0 => y = 7

Plugging y = 7 in the first equation yields:

x - 7+ 1 = 0 => x = 6

The coordinates of the point P are (6;7).

The line passing through P is 7 = 6m + b.

The point (3,2) is at the distance d =   from the line 7 = 6m + b , hence you may replace these values in the following formula:

 

The equation of the line passing through P and found at a distance of  from the point (3;2) is 6m + b - 7 = 0, m and b being in the following relation 
 

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Please find this answer

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Required expression 8x-9y+11=0

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