CUET UG 2022 — Mathematics Applied-Mathematics

When two matrices are equal, their corresponding entries (same row, same column) must be equal.

$$\begin{bmatrix} 5x + 8 & 7 \\ y + 3 & 10x + 12 \end{bmatrix} = \begin{bmatrix} 2 & 3y + 1 \\ 5 & 0 \end{bmatrix}$$[5x+8y+3​710x+12​]=[25​3y+10​]

This gives us four equations:

\5x + 8 = 2 \quad\cdots(1)$\$7 = 3y + 1 \quad\cdots(2)

$$y + 3 = 5 \quad\cdots(3)$$y+3=5⋯(3)

\10x + 12 = 0 \quad\cdots(4)$

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From equation $(1)$(1):

\5x + 8 = 2$\$5x = -6

x = -\dfrac{6}{5}$ Verifying with equation $(4)$: $\$10\left(-\dfrac{6}{5}\right) + 12 = -12 + 12 = 0 \;\checkmark$ <hr> From equation $(3)$:y + 3 = 5$$$$y = 2$

Verifying with equation $(2)$(2):

\3(2) + 1 = 7 ;\checkmark$

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Now substituting $x = -\dfrac{6}{5}$x=−56​ and $y = 2$y=2 into $5x + 3y$5x+3y:

\5x + 3y = 5\left(-\dfrac{6}{5}\right) + 3(2)$$$$= -6 + 6$$$$= \boxed{0}$$