Question

Find the equation of the line passing through the point (-1,2)

and the points of intersections of the line 2x - 3y + 11 = 0 and
5x + y + 3 = 0​

Answer 1

`y=-5x-3`

Explanation:

Hi there!

What we need to know:

Linear equations are typically organized in slope-intercept form:
`y=mx+b` where m is the slope and b is the y-intercept (the value of y when x is 0).

To solve for the equation of the line, we would need to:

1. Find the point of intersection between the two given lines
2. Use the point of intersection and the given point (-1,2) to solve for the slope of the line
3. Use a point and the slope in
   `y=mx+b` to solve for the y-intercept
4. Plug the slope and the y-intercept back into
   `y=mx+b` to achieve the final equation

1) Find the point of intersection between the two given lines

`2x - 3y + 11 = 0`

`5x + y + 3 = 0`

Isolate y in the second equation:

`y=-5x-3`

Plug y into the first equation:

`2x - 3(-5x-3) + 11 = 0\\2x +15x+9 + 11 = 0\\17x+20 = 0\\17x =-20\\\\x=\displaystyle-(20)/(17)`

Plug x into the second equation to solve for y:

`5x + y + 3 = 0\\\\5(\displaystyle-(20)/(17)) + y + 3 = 0\\\\\displaystyle-(100)/(17) + y + 3 = 0`

Isolate y:

`y = -3+\displaystyle(100)/(17)\\y = (49)/(17)`

Therefore, the point of intersection between the two given lines is
`(\displaystyle-(20)/(17),(49)/(17))`.

2) Determine the slope (m)

`m=\displaystyle (y_2-y_1)/(x_2-x_1)` where two points that fall on the line are
`(x_1,y_1)` and
`(x_2,y_2)`

Plug in the two points
`(\displaystyle-(20)/(17),(49)/(17))` and (-1,2):

`m=\displaystyle (\displaystyle(49)/(17)-2)/(\displaystyle-(20)/(17)-(-1))\\\\\\m=\displaystyle (\displaystyle(15)/(17) )/(\displaystyle-(20)/(17)+1)\\\\\\m=\displaystyle (\displaystyle(15)/(17) )/(\displaystyle-(3)/(17) )\\\\\\m=-5`

Therefore, the slope of the line is -5. Plug this into
`y=mx+b`:

`y=-5x+b`

2) Determine the y-intercept (b)

`y=-5x+b`

Plug in the point (-1,2) and solve for b:

`2=-5(-1)+b\\2=5+b\\-3=b`

Therefore, the y-intercept is -3. Plug this back into
`y=-5x+b`:

`y=-5x+(-3)\\y=-5x-3`

I hope this helps!