Question

How do you find the values of x and y given $\left[ \begin{matrix} 4x-3y \\\ x+y \\\ \end{matrix} \right]=\left[ \begin{matrix} 11 \\\ 1 \\\ \end{matrix} \right]$?

Answer 1

We define the meaning of equality of two matrices. From the given relation of $\left[ \begin{matrix} 4x-3y \\\ x+y \\\ \end{matrix} \right]=\left[ \begin{matrix} 11 \\\ 1 \\\ \end{matrix} \right]$, we equate their corresponding elements. We get two equations of two unknowns. We solve them to get the values of x and y.

Complete step-by-step solution:
In the given relation we have been given equality of two matrices.
Let’s assume that A and B are two matrices where $A=\left[ {{a}_{ij}} \right]$ and $B=\left[ {{b}_{ij}} \right]$. Here ${{a}_{ij}}$ and ${{b}_{ij}}$ are the ${{i}^{th}}$ row and ${{j}^{th}}$ column elements of the matrices A and B respectively.
If both the matrices are of the same order then the equality $A=B$ gives us ${{a}_{ij}}={{b}_{ij}}$.
This means equality of two same order matrices gives equality of corresponding elements of those two matrices.
For our given relation of $\left[ \begin{matrix} 4x-3y \\\ x+y \\\ \end{matrix} \right]=\left[ \begin{matrix} 11 \\\ 1 \\\ \end{matrix} \right]$, let’s assume $M=\left[ \begin{matrix} 4x-3y \\\ x+y \\\ \end{matrix} \right];N=\left[ \begin{matrix} 11 \\\ 1 \\\ \end{matrix} \right]$.
Both matrices are of order $\left( 2\times 1 \right)$.
That’s why we can equalise corresponding elements of those matrices.
$\left[ \begin{matrix} 4x-3y \\\ x+y \\\ \end{matrix} \right]=\left[ \begin{matrix} 11 \\\ 1 \\\ \end{matrix} \right]$ gives $4x-3y=11....(i)$ and $x+y=1.....(ii)$.
We have two unknowns and two equations to solve.
We multiply 3 with the equation (ii) and add that to equation (i).
We get $3\left( x+y \right)=3\Rightarrow 3x+3y=3.....(iii)$
Now adding $\left( 4x-3y \right)+\left( 3x+3y \right)=11+3$. Simplifying we get
$\begin{aligned} & 7x=14 \\\ & \Rightarrow x=\dfrac{14}{7}=2 \\\ \end{aligned}$
From the value of x, we get $y=1-x=1-2=-1$.
Therefore, the values of x and y is $x=2,y=-1$.

Note: Order mismatch of matrices can’t solve the equality of matrices. The corresponding elements can’t be projected with one another. The equality of $\left( m\times n \right)$ ordered matrix with $\left( n\times m \right)$ ordered matrix is also not possible.