Linear Equations Question

Question

I am stuck trying to solve the following problem:

"Find the value of 'c' if the line through $(7, 2c-5) ,(3,-2)$ is perpendicular to the line through $(-6, c+4) and (-9,5)$."

The answer I have found is that there are "no real solutions".

Any help is appreciated.

Edit: This is how I approached the problem:

I knew that the slope of a line is basically "change in y / change in x", and therefore $y1-y2 / x1-x2$. Additionally, the slopes of the lines are also opposite reciprocals. Using that logic I created the equation $2c-5+2/7-3 = 3/-c+1$. I cross-multiplied and got a quadratic equation with no solution.

Answer 1

Your answer is correct, but the computations are a bit simpler if you use directing vectors instead of slopes, since the latter are not given:

Directing vectors are, respectively $\;\vec u=(4, 2c-3)$ and $\vec v=(3, c-1)$. The lines are perpendicular if and only if $\;\vec u\cdot\vec v=0$.

This equation yields instantly the same equation as your final equation.

Answer 2

For the slopes in this case must hold:$$m_1\cdot m_2=-1$$ so you have to solve $$\frac{3-2c}{-4}\times\frac{1-c}{-3}=-1$$