Question

If the pair of linear equations

a1x+b1y+c1=0

a2x+b2y+c2=0

has a unique solution, then

Answer 1

Answer:

Let's not use the standard bookish formula i.e. $\frac{a_1}{a_2}\neq \frac{b_1}{b_2}\neq \frac{c_1}{c_2}$

First, let's see what does unique solution of two linear equation means graphically.

See the attached graph.

Step-by-step explanation:

It means that two lines are intersecting at only one point.

Now think, what does the bookish formula represents?

The formula represents the ratio of the coefficients, and what does coefficients do. They determine the slope of a line.

If you really want to understand, see the reasoning below:

We can write the standard form of equation of a line as:

$y=mx+c$

where m = slope of the line

Now the given pair of linear equations can also be represented as above:

$y=-\frac{a_1}{b_1}x-\frac{c_1}{b_1} \\\\y=-\frac{a_2}{b_2}x-\frac{c_2}{b_2}$

We just did a bit of rearrangement.

Comparing with the standard form, let

$m_1=-\frac{a_1}{b_1}\\\\m_2=-\frac{a_2}{b_2}$

And, we know, the slope of two lines are equal if they are parallel.

If they are parallel, then $m_1=m_2$, they either have infinitely many solutions or have no solution at all. (Out of scope for this question)

If they are not parallel, then $m1\neq m_2$, they have to intersect each other or we can say it has an unique solution. (Our primary focus)

Then,

$-\frac{a_1}{b_1}\neq -\frac{a_2}{b_2}\\ \\ or \ \frac{a_1}{a_2}\neq \frac{b_1}{b_2}$

So, now you can see why the bookish formula makes sense.

Additional info: Linear equation in two variables means the graph is 2 dimensional and represents a line. When you add another variable (e.g. ax+by+cz+d=0), the graph is 3 dimensional, and represents a plane.