Find the equation of the line that passes through the point of intersection of $3x-5y+10=0$ and $2x+3y=6$ and also passes through the point $(-2,0)$.
I have an idea on how to do this but i'm not sure if I'm right
1) Solve the two equations by elimination/substitution (find $x$ and $y$ value)
2) After finding the $x$ and $y$ value from the two equations pair it alongside $(-2,0)$.
3) Then find the equation from those two set of points
Is this right, if not can someone describe to me what to do?
Best Answer
That’s a solid approach, but there’s a quicker way.
Every line that passes through the intersection point has an equation that’s a linear combination of the equations of the two lines, i.e., $\lambda(3x-5y+10)+\mu(2x+3y-6)=0$. We want this line to pass through $(-2,0)$, so after substituting these coordinates into the combined equation, the problem reduces to solving $4\lambda=10\mu$ with $\lambda$ and $\mu$ not both zero. One solution is $\lambda=5$, $\mu=2$; substitute into the combined equation and simplify. Note that neither of the given lines passes through $(-2,0)$, so you can reduce the number of unknowns by setting either $\lambda$ or $\mu$ to $1$ right off the bat. Doing so eliminates the corresponding line as a potential solution, but we’ve already verified that it can’t be the solution to the problem.
Going even further, you can use a technique known as Plücker’s mu to write a solution down directly, without solving any equations whatsoever. It is $$(2\cdot(-2)+3\cdot0-6)(3x-5y+10)-(3\cdot(-2)-5\cdot0-6)(2x+3y-6)=0,$$ which simplifies to $x-y+2=0$.