Real Numebrs Chapter 1 class 10 Basic Concepts
1. Two linear equations in the same two variables are called a pair of linear equations in two variables. The most general form of a pair of linear equations is
$$
\begin{aligned}
& a_{1} x+b_{1} y+c_{1}=0 \\
& a_{2} x+b_{2} y+c_{2}=0
\end{aligned}
$$
where $a_{1}, a_{2}, b_{1}, b_{2}, c_{1}, c_{2}$ are real numbers, such that $a_{1}^{2}+b_{1}^{2} \neq 0, a_{2}^{2}+b_{2}^{2} \neq 0$.
2. Graphical Method to solve a pair of linear equations in two varibale:
The graph of a pair of linear equations in two variables is represented by two lines.
(i) If the lines intersect at a point, then that point gives the unique solution of the two equations. In this case, the pair of equations is consistent.
(ii) If the lines coincide, then there are infinitely many solutions - each point on the line being a solution. In this case, the pair of equations is dependent (consistent).
(iii) If the lines are parallel, then the pair of equations has no solution. In this case, the pair of equations is inconsistent.
3. If a pair of linear equations is given by $a_{1} x+b_{1} y+c_{1}=0$ and $a_{2} x+b_{2} y+c_{2}=0$, then the following situations can arise :
(i) $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{1}}:$ In this case, the pair of linear equations is consistent.
(ii) $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}:$ In this case, the pair of linear equations is inconsistent.
(iii) $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$ : In this case, the pair of linear equations is dependent and consistent.