Question

question number 14 16 17 and 18 please give me the answer of these questions

Answer 1

$\textbf{ \huge{ Hello Mate! }}$

14.

$\frac{3x + 2}{7} + \frac{4(x + 1)}{5} = \frac{2(2x + 1)}{3} \\ \frac{3x + 2}{7} + \frac{4x + 4}{5} = \frac{4x + 2}{3} \\ \frac{3x + 2}{7} \times 105 + \frac{4x + 4}{5} \times 105 = \frac{4x + 2}{3} \times 105 \\ 15(3x + 2) + 21(4x + 4) = 35(4x + 2) \\ 45x + 30 + 84x + 84 = 140x + 70 \\ 129x + 114 = 140x + 70 \\ 44 = 11x \\ x = 4$

Note: 105 is L.C.M of 7,3 and 5

$\boxed{ \textsf{ \red{ Answer\:is\:4 }}}$

15.

2x - 5y = 10

Keeping value of x = 0

2(0) - 5y = 10
y = - 2

Keeping y = 0

2x - 5(0) = 10
x = 5

Keeping x = 1

2(1) - 5y = 10
- 5y = 10 - 2
y = 8 / - 5 = - 1.6

$\boxed{ \textsf{ \red{ Solutions\:are\:(0,-2);(5,0)\:and\:(1,-1.6) }}}$

Note: Answers can be different.

16.

L.H.S =>$\frac{1}{2 + \sqrt{3} } = \frac{1}{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} } \\ = \frac{2 - \sqrt{3} }{ {2}^{2} - { \sqrt{3} }^{2} } = \frac{2 - \sqrt{3} }{4 - 3} \\ = 2 - \sqrt{3} \\ \frac{2}{ \sqrt{5} - \sqrt{3} } = \frac{2}{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} + \sqrt{3} } \\ = \frac{2( \sqrt{5} + \sqrt{3} )}{2} = \sqrt{5} + \sqrt{3} \\ \frac{1}{2 - \sqrt{5} } = \frac{1}{2 - \sqrt{5} } \times \frac{2 + \sqrt{5} }{2 + \sqrt{5} } \\ \frac{2 + \sqrt{5} }{ 4 - 5} = - 2 - \sqrt{5} \\ 2 - \sqrt{3} + \sqrt{5} + \sqrt{3} - 2 - \sqrt{5} \\ = 0$ => R.H.S

$\boxed{ \textsf{ \red{ Hence\:Proved }}}$

17.

$a - b = 7$

Squaring both sides we get,

${(a - b)}^{2} = {7}^{2} \\ {a}^{2} + {b}^{2} - 2ab = 49 \\ 85 - 2ab = 49 \\ - 2ab = 49 - 85 \\ ab = \frac{ - 36}{ - 2} = 18 \\ {a}^{3} - {b}^{3} = ( a - b)( {a}^{2} + {b}^{2} + ab) \\ = 7(85 + 18) \\ = 7 \times 103 = 721$

$\boxed{ \textsf{ \red{ Answer\:is\: 721 }}}$

18.

$x + \frac{1}{x} = 3$
On squaring both sides we get,

${(x + \frac{1}{x} )}^{2} = {3}^{2} \\ {x}^{2} + \frac{1}{ {x}^{2} } + 2 \times x \times \frac{1}{x} = 9 \\ {x}^{2} + \frac{1}{ {x}^{2} } = 9 - 2 = 7$

On again squaring we get,

${( {x}^{2} + \frac{1}{ {x}^{2} } )}^{2} = {7}^{2} \\ {x}^{4} + \frac{1}{ {x}^{4} } + 2 \times {x}^{2} \times \frac{1 }{ {x}^{2} } = 49 \\ {x}^{4} + \frac{1}{ {x}^{4} } = 49 - 2 = 47$

$\boxed{ \textsf{ \red{ Answer\:is\:47 }}}$

$\textbf{ Have fantastic future ahead! }$