Question

Examine the consistency of the system of linear equations $x + 2y = 2$ and
$2x + 3y = 3$.

Answer 1

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:

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.