Question

Examine the consistency of the system of linear equations \[x + 2y = 2\] and
\[2x + 3y = 3\].

Answer 1

Hint: We will be solving this question by writing the given equations into the matrix form
\[{\text{A}}x = {\text{B}}\].
For example, we have two equations
\[x + y = 1\] and
\[3x + y = 2\], so we will write these equations into matrix form
\[{\text{A}}x = {\text{B}}\] by using the coefficients of each equation to form each row of the matrix as shown below:
\[\left[ {\begin{array}{*{20}{c}}
  1&1 \\
  3&1
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  x \\
  y
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  1 \\
  2
\end{array}} \right]\].

Complete step-by-step solution:
Step 1: For checking the consistency of the system of linear equations, first of all, we will write the equations into the matrix form
\[{\text{A}}x = {\text{B}}\] as shown below:
\[ \Rightarrow \left[ {\begin{array}{*{20}{c}}
  1&2 \\
  2&3
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  x \\
  y
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  2 \\
  3
\end{array}} \right]\]
Where,
\[{\text{A = }}\left[ {\begin{array}{*{20}{c}}
  1&2 \\
  2&3
\end{array}} \right]\],

\[{\text{X = }}\left[ {\begin{array}{*{20}{c}}
  x \\
  y
\end{array}} \right]\] and \[{\text{B = }}\left[ {\begin{array}{*{20}{c}}
  2 \\
  3
\end{array}} \right]\].
Step 2: Now we will be calculating the determinant of the matrix \[{\text{A}}\]by using the formula \[\left| {\text{A}} \right| = ad - bc\], where \[a\],\[b\] are the elements of the first row and \[b\],\[d\] are the elements of the second one in any \[2 \times 2\] matrix.
By using the formula of determinant in \[{\text{A = }}\left[ {\begin{array}{*{20}{c}}
  1&2 \\
  2&3
\end{array}} \right]\], we get:
\[ \Rightarrow \left| {\text{A}} \right| = \left( {1 \times 3} \right) - \left( {2 \times 2} \right)\]
By solving the brackets in the above expression, we get:
\[ \Rightarrow \left| {\text{A}} \right| = 3 - 4\]
By doing the final subtraction in the RHS side of the above expression, we get:
\[ \Rightarrow \left| {\text{A}} \right| = - 1\]
Which means that \[\left| {\text{A}} \right| \ne 0\]. So, if the determinant of the linear equations is not equal to zero then consistency exists.
Therefore, we can say that the system of equations is consistent.

The system of equations is consistent.

Note: Students need to remember the formula for calculating the determinant of the matrix which is denoted by the symbol \[\left| {\text{A}} \right|\]. So, for any \[2 \times 2\] matrix \[{\text{A = }}\left[ {\begin{array}{*{20}{c}}
  a&b \\
  c&d
\end{array}} \right]\], the determinant of the matrix will be equals to as below:
\[\left| {\text{A}} \right| = ad - bc\].
Also, you should remember that if the determinant of the matrix equals zero then the system of equations may be either consistent or inconsistent but if the determinant is non-zero then the system of equations is always consistent.