If [(x+3,4)(y-4,x+y)]= [(5,4)(3,9],find the value of x and y. ​

Question

If \(\begin{bmatrix} x+3& 4 \\[0.3em] y-4 & x+y \\[0.3em] \end{bmatrix}\)= \(\begin{bmatrix} 5& 4 \\[0.3em] 3 & 9 \\[0.3em] \end{bmatrix}\), find the value of x and y.

Answer 1

We are given with,

\(\begin{bmatrix} x+3& 4 \\[0.3em] y-4 & x+y \\[0.3em] \end{bmatrix}\)= \(\begin{bmatrix} 5& 4 \\[0.3em] 3 & 9 \\[0.3em] \end{bmatrix}\)

We need to find the values of x and y. 

We know by the property of matrices,

 \(\begin{bmatrix} a_{11}& a_{12} \\[0.3em] a_{21} & a_{22} \\[0.3em] \end{bmatrix}\)= \(\begin{bmatrix} b_{11}& b_{12} \\[0.3em] b_{21} & b_{22} \\[0.3em] \end{bmatrix}\)

This implies, 

a₁₁ = b₁₁, 

a₁₂ = b₁₂, 

a₂₁ = b₂₁ and

 a₂₂ = b₂₂ 

So, if we have

 \(\begin{bmatrix} x+3& 4 \\[0.3em] y-4 & x+y \\[0.3em] \end{bmatrix}\)= \(\begin{bmatrix} 5& 4 \\[0.3em] 3 & 9 \\[0.3em] \end{bmatrix}\)

Corresponding elements of two matrices are equal. 

That is,

x + 3 = 5 …(i) 

4 = 4 

y – 4 = 3 …(ii) 

x + y = 9 …(iii)

To solve for x and y, 

We have three equations (i), (ii) and (iii).

From equation (i), 

x + 3 = 5 

⇒ x = 5 – 3 

⇒ x = 2

From equation (ii), 

y – 4 = 3 

⇒ y = 3 + 4 

⇒ y = 7 

We need not solve equation (iii) as we have got the values of x and y,

Thus, 

The values of x = 2 and y = 7.