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(A) Let the two-digit number be $10x + y$,where $x$ is the tens digit and $y$ is the units digit.Case $I$: Multiplying the sum of the digits by $8$ an

Vedclass · Accepted Answer

$83$

Vedclass · Answer

(A) Let the two-digit number be $10x + y$,where $x$ is the tens digit and $y$ is the units digit.Case $I$: Multiplying the sum of the digits by $8$ and then subtracting $5$ gives the number:$8(x + y) - 5 = 10x + y$$8x + 8y - 5 = 10x + y$$2x - 7y = -5$ ... $(i)$Case $II$: Multiplying the difference of the digits by $16$ and then adding $3$ gives the number:$16(x - y) + 3 = 10x + y$$16x - 16y + 3 = 10x + y$$6x - 17y = -3$ ... $(ii)$To solve the system,multiply equation $(i)$ by $3$:$6x - 21y = -15$ ... $(iii)$Subtract equation $(iii)$ from equation $(ii)$:$(6x - 17y) - (6x - 21y) = -3 - (-15)$$4y = 12$$y = 3$Substitute $y = 3$ into equation $(i)$:$2x - 7(3) = -5$$2x - 21 = -5$$2x = 16$$x = 8$The number is $10x + y = 10(8) + 3 = 83$.

$A$ two-digit number is obtained by either multiplying the sum of the digits by $8$ and then subtracting $5$,or by multiplying the difference of the digits by $16$ and then adding $3$. Find the number.

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