Question

What is the formula of a unique solution?

Answer 1

Hint: This question is based on the linear equation. An equation of the form \[ax + by + c = 0\] where \[x,y\] are variables and \[a,b,c \ne 0\] are constants is called a linear equation. It is called a linear equation because the power of the variables is \[1\], that is, the linear equations are of the first order and the graph plot by the value of \[x,y\] will be a straight line. In this question we are asked to state the formula of a unique solution and the formula is \[\dfrac{a}{p} \ne \dfrac{b}{q}\]. Let us prove the formula of a unique solution.

Complete step-by-step answer:
In this question we are asked to state the formula of a unique solution. Before that let us see what the unique solution is. For that let us consider two linear equations: \[ax + by + c = 0\] and \[px + qy + r = 0\] where \[a,b,c,p,q,r\] are constants and \[x,y\] are variables. By plotting the graph, the two lines represented by the equations \[ax + by + c = 0\] and \[px + qy + r = 0\] intersect at a point then the given equations will have unique solutions.
The formula of the unique solution is \[\dfrac{a}{p} \ne \dfrac{b}{q}\].
It means that the slope of the two lines should be different for a unique solution.
Let's verify it by expressing both the equation in terms of \[y\] because slope intercept formula \[y = mx + b\] where \[m\] is the slope of the line and \[b\] is the \[y\] intercept.
\[ax + by + c = 0\]
Changing it into terms of \[y\], shift everything to the R.H.S. except \[y\].
\[by + c = - ax\]
\[by = - ax - c\]
While moving the coefficient of \[y\] to R.H.S. it will become as denominator,
\[y = \dfrac{{ - ax - c}}{b}\]
Here the denominator \[b\] is common let split and write,
\[\therefore y = - \dfrac{a}{b}x - \dfrac{c}{a}\] ……………………………. (1)
Similarly,
\[px + qy + r = 0\]
\[y = - \dfrac{p}{q}x - \dfrac{r}{q}\] ……………………………….. (2)
From equation \[1\] & \[2\] for a unique solution, the slope of the two lines should be different.
Slopes of the lines \[ \Rightarrow - \dfrac{a}{b} \ne - \dfrac{p}{q}\].
Here both the sides have \[ - \] we cannot simply cancel it because they are not equal. To cancel the \[ - \] the equation will become \[\dfrac{a}{p} \ne \dfrac{b}{q}\].
Hence, the formula of the unique solution is \[\dfrac{a}{p} \ne \dfrac{b}{q}\].

Note: Slope intercept formula \[y = mx + b\].
For example: \[y = 5x + 9\].
As per the universal truth everything has two faces. If there is a unique solution then there must be a non-unique solution also. The slope of the two lines should be the same for a non-unique solution. The formula of a non-unique solution is \[\dfrac{a}{p} = \dfrac{b}{q}\].