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{aligned}
& a_1 x+b_1 y+c_1=0 \\
& a_2 x+b_2 y+c_2=0
\end{aligned}
$
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{aligned}
& 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{aligned}
$
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{aligned}
& a_1 x+b_{1 y}=c_1 \\
& a_2 x+b_2 y=c_2
\end{aligned}
$
It has exactly one solution if

$
\frac{\mathrm{a}_1}{\mathrm{a}_2} \neq \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} \neq \frac{\mathrm{c}_1}{\mathrm{c}_2}
$
For example, $x+y=5$ and $2 x+2 y=5$ 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.