Question

Hello frnds ❤️

find the equation of a straight line passing through the point of intersection of two lines 5x-3Y-1=0, 2x+ 3y- 23=0 and it is perpendicular to the line 5x-3y+1=2

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Answer 1

Answer:

3x + 5y = 91

Step-by-step explanation:

Given Problem:

Find the equation of a straight line passing through the point of intersection of two lines 5x-3Y-1=0, 2x+ 3y- 23=0 and it is perpendicular to the line 5x-3y+1=2

Solution:

First,We have to find out intersecting point of given lines 5x -3y = 1 and 2x + 3y = 23.  

5x - 3y = 1..................Equation(1)

2x + 3y = 23.............Equation(2)

Add equations:

Equation(1) and (2) ,  

7x = 24 ⇒ x = $\frac{24}{7}$,Put it in Equation(1)  

3y = $\frac{120}{7}$ -1 ⇒y = $\frac{113}{21}$

Hence,

Unknown line is passing through the point ($\frac{24}{7}$ ,$\frac{113}{21}$)

Now,

Unknown line is perpendicular upon 5x - 3y = 1

Means, slope of unknown line × { slope of 5x - 3y = 1} = -1

Let slope of unknown line is x  

x × $\frac{5}{3}$ = -1 ⇒ x = $\frac{-3}{5}$

Now,

Equation of unknown line is ,

(y - $\frac{113}{7}$) = $\frac{-3}{5}$(x - $\frac{24}{7}$)  

⇒5y - $\frac{565}{7}$ + 3x - $\frac{72}{7}$ = 0

⇒ 3x + 5y - 91= 0

Hence,

Answer is 3x + 5y = 91

Answer 2

$\huge\mathfrak\red{Answer}$

POINT OF INTERSECTION

$5x - 3y - 1 = 0 \\ 2x + 3y - 23 = 0 \\ \\ add \: both \: \\ \\ 7x - 24 = 0 \\ 7x = 24 \\ x = \frac{24}{7} \\ \\ put \: value \: in \: one \: of \: the \: equation \\ \\ 2( \frac{24}{7} ) + 3y - 23 = 0 \\ \\ \frac{48}{7} + 3y - 23 = 0 \\ \\ \frac{48 - 161}{7} + 3y = 0 \\ \\ \frac{ - 113}{7} + 3y = 0 \\ \\ 3y = \frac{113}{7} \\ \\ y = \frac{113}{21} \\ \\ eq \: of \: line \: perpendicular \: to \\ 5x - 3y - 1 = 0 \: is \\ 3x + 5y + \gamma = 0 \: where \: \gamma \\ is \: an \: arbitrary \: constant \\ \\ this \: eq \: passe \: through \: the \: intersection \\ point \\ \\ 3( \frac{24}{7} ) + 5( \frac{113}{21} ) + \gamma = 0 \\ \frac{21 \times 24 + 565}{21} + \gamma = 0 \\ \\ \frac{1069}{21} = - \gamma \\ \\ required \: equation \\ \\ \\ 3x + 5y - \frac{1069}{21} = 0$

EQUATION=3x + 5y - 91=0