System of Linear Equations - Practice Questions & MCQ

System of Linear Equation

1. System of 2 Linear Equations:

It is a pair of linear equations in two variables. It is usually of the form

$\begin{array}{rl}& a_{1}x+b_{1}y+c_1=0\\ & a_{2}x+b_{2}y+c_2=0\end{array}$
Finding a solution for this system means to find the values of x and y that satisfy both the equations.
2. System of 3 Linear Equations:

It is a group of 3 linear equations in three variables. It is usually of the form

$\begin{array}{rl}& a_{1}x+b_{1}y++c_{1}z+d_1=0\\ & a_{2}x+b_{2}y++c_{2}z+d_2=0\\ & a_{3}x+b_{3}y++c_{3}z+d_3=0\end{array}$
Finding a solution for this system means to find the values of $x,y$ and $z$ that satisfy all the three equations.

The system of equations are broadly classified into two types:

Consistent system of equations:

A system of equations is said to be consistent if it has at least one solution. Let the given system of equation is

$\begin{array}{rl}& a_{1}x+b_{1y}=c_1\\ & a_{2}x+b_{2}y=c_2\end{array}$
It has exactly one solution if

$\frac{{\mathrm{a}}_1}{{\mathrm{a}}_2}\ne \frac{{\mathrm{b}}_1}{{\mathrm{b}}_2}$

E.g., $x+y=2$
$\mathrm{x}-\mathrm{y}=6$ is consistent because it has a solution $\mathrm{x}=4$ and $\mathrm{y}=-2$.
Given lines are non-parallel, hence lines will have one point of intersection.
It has infinite solutions if

$\frac{{\mathrm{a}}_1}{{\mathrm{a}}_2}=\frac{{\mathrm{b}}_1}{{\mathrm{b}}_2}=\frac{{\mathrm{c}}_1}{{\mathrm{c}}_2}$

In this case two lines represented by these lines coincide, so there are infinite pair of values of x and y that satisfy both the equations. This case is also counted as consistent as there is at least one solution.

Inconsistent equation:

A system of equations is said to be inconsistent if it has no solution.
Let $a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$, then equation are inconsistent and has no solution if

$\frac{{\mathrm{a}}_1}{{\mathrm{a}}_2}=\frac{{\mathrm{b}}_1}{{\mathrm{b}}_2}\ne \frac{{\mathrm{c}}_1}{{\mathrm{c}}_2}$
For example, $x+y=5$ and $2x+2y=5$ is inconsistent as it has no solution, just by seeing the equation, we get that it is the equation of two different parallel lines which never intersect. This lines are non-intersecting, hence there is no solution to this system.