Equation of line passing through point of intersection of the two lines and having slope calculator

SolutionMethods

1. Find the equation of a line passing through the point of intersection of lines

x-4y+18=0 $x-4y+18=0$ and

x+y-12=0 $x+y-12=0$ and having slope

2 $2$

2. Find the equation of a line passing through the point of intersection of lines

2x+3y+4=0 $2x+3y+4=0$ and

3x+6y-8=0 $3x+6y-8=0$ and having slope

2 $2$

3. Find the equation of a line passing through the point of intersection of lines

x=3y $x=3y$ and

3x=2y+7 $3x=2y+7$ and having slope

-12 $-1/2$

Example

1. Find the equation of a line passing through the point of intersection of lines x-4y+18=0 $x-4y+18=0$ and x+y-12=0 $x+y-12=0$ and having slope 2 $2$

Solution:
The point of intersection of the lines can be obtainted by solving the given equations

x-4y+18=0 $x-4y+18=0$

$:.x-4y=-18$

and

$x+y-12=0$

$:.x+y=12$

$x-4y=-18 ->(1)$

$x+y=12 ->(2)$

Substracting

$=>-5y=-30$

$=>5y=30$

$=>y=30/5$

$=>y=6$

Putting

$y=6$ in equation

$(2)$ , we have

$x+6=12$

$=>x=12-6$

$=>x=6$

$:.x=6" and "y=6$

$:. (6,6)$ is the intersection point of the given two lines.

The equation of a line with slope m and passing through

$(x_1,y_1)$ is

$y-y_1=m(x-x_1)$

Here Point

$(x_1,y_1)=(6,6)$ and Slope

$m=2$ (given)

$:. y-6=2(x-6)$

$:. y -6=2x -12$

$:. 2x-y-6=0$

Hence, The equation of line is

$2x-y-6=0$

2. Find the equation of a line passing through the point of intersection of lines $2x+3y+4=0$ and $3x+6y-8=0$ and having slope $2$

Solution:
The point of intersection of the lines can be obtainted by solving the given equations

$2x+3y+4=0$

$:.2x+3y=-4$

and

$3x+6y-8=0$

$:.3x+6y=8$

$2x+3y=-4 ->(1)$

$3x+6y=8 ->(2)$

equation

$(1) xx 3 =>6x+9y=-12$

equation

$(2) xx 2 =>6x+12y=16$

Substracting

$=>-3y=-28$

$=>3y=28$

$=>y=28/3$

Putting

$y=28/3$ in equation

$(1)$ , we have

$2x+3(28/3)=-4$

$=>2x=-4-28$

$=>2x=-32$

$=>x=-16$

$:.x=-16" and "y=28/3$

$:. (-16,28/3)$ is the intersection point of the given two lines.

The equation of a line with slope m and passing through

$(x_1,y_1)$ is

$y-y_1=m(x-x_1)$

Here Point

$(x_1,y_1)=(-16,28/3)$ and Slope

$m=2$ (given)

$:. y-28/3=2(x+16)$

$:. y -28/3=2x +32$

$:. 2x-y+124/3=0$

$:. 6x-3y+124=0$

Hence, The equation of line is

$6x-3y+124=0$

3. Find the equation of a line passing through the point of intersection of lines $x=3y$ and $3x=2y+7$ and having slope $-1/2$

Solution:
The point of intersection of the lines can be obtainted by solving the given equations

$x=3y$

$:.x-3y=0$

and

$3x=2y+7$

$:.3x-2y=7$

$x-3y=0 ->(1)$

$3x-2y=7 ->(2)$

equation

$(1) xx 3 =>3x-9y=0$

equation

$(2) xx 1 =>3x-2y=7$

Substracting

$=>-7y=-7$

$=>7y=7$

$=>y=7/7$

$=>y=1$

Putting

$y=1$ in equation

$(2)$ , we have

$3x-2(1)=7$

$=>3x=7+2$

$=>3x=9$

$=>x=3$

$:.x=3" and "y=1$

$:. (3,1)$ is the intersection point of the given two lines.

The equation of a line with slope m and passing through

$(x_1,y_1)$ is

$y-y_1=m(x-x_1)$

Here Point

$(x_1,y_1)=(3,1)$ and Slope

$m=-1/2$ (given)

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

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

$:. 2y -2=-x +3$

$:. x+2y-5=0$

Hence, The equation of line is

$x+2y-5=0$