the sum of two digit number …

Let in the first two digit number,
Unit's digit = x
And, ten's digit = y
Them, the number = 10y + x
On interchanging the digits, in the new number unit's digit = y
And, ten's digit = x
{tex}\therefore{/tex} The new number = 10x + y
According to the question,
(10y + x) + (10x + y) = 110
{tex}\Rightarrow{/tex} 11x + 11y = 110
{tex}\Rightarrow{/tex} x + y = 10 ....Dividing throughout by 11
{tex}\Rightarrow{/tex} x + y - 10 = 0 ....(1)
And (10y + x) - 10 = 5(x + y) + 4
{tex}\Rightarrow{/tex} 10y + x - 10 = 5x + 5y + 4
{tex}\Rightarrow{/tex} -4x +5y  -14 =0
{tex}\Rightarrow{/tex} 4x - 5y + 14 = 0 ....(2)
To solve the equation (1) and (2) by cross multiplication method, we draw the diagram below;

Then,
{tex}\Rightarrow \;\frac{x}{{(1)(14) - ( - 5)( - 10)}} = \frac{y}{{( - 10)(4) - (14)(1)}}{/tex}{tex}= \frac{1}{{(1)( - 5) - (4)(1)}}{/tex}
{tex}\Rightarrow \;\frac{x}{{14 - 50}} = \frac{y}{{ - 40 - 14}} = \frac{1}{{ - 5 - 4}}{/tex}
{tex}\Rightarrow \;\frac{x}{{ - 36}} = \frac{y}{{ - 54}} = \frac{1}{{ - 9}}{/tex}
{tex}\Rightarrow \;x = \frac{{ - 36}}{{ - 9}} = 4{/tex} and {tex}\Rightarrow \;y = \frac{{ - 54}}{{ - 9}} = 6{/tex}
Hence, the first two digit number = 10 {tex}\times{/tex} 6 + 4 = 60 + 4 = 64
Verification. Substituting x = 4, y = 6, we find that both the equations (1) and (2) are
satisfied as shown below:
x + y - 10 = 4 + 6 - 10 = 0
4x - 5y + 14 = 4(4) - 5(6) + 14
= 16 - 30 + 14 = 0
Hence, the solution we have got is correct.