Question

find the gradient of the line passing through 3,-2 and through the point of intersection of the line 3x-5y=6 and x+y+3=0

ans= 5/27

but how?​

Answer 1

Given Question :-

Find the gradient of the line passing through (3,-2) and through the point of intersection of the line 3x-5y=6 and x+y+3=0.

$\large\underline{\sf{Solution-}}$

Let first find the point of intersection of the lines

$\rm :\longmapsto\:3x - 5y = 6 - - - (1)$

and

$\rm :\longmapsto\:x + y = - 3 - - - (2)$

Multiply equation (2) by 5, we get

$\rm :\longmapsto\:5x + 5y = - 15 - - - (3)$

On adding equation (1) and (3), we get

$\rm :\longmapsto\:8x = - 9$

$\bf\implies \:x = - \dfrac{9}{8}$

On substituting the value of x in equation (2), we get

$\rm :\longmapsto\: - \dfrac{9}{8} + y = - 3$

$\rm :\longmapsto\: y = - 3 + \dfrac{9}{8}$

$\rm :\longmapsto\: y = \dfrac{ - 24 + 9}{8}$

$\bf\implies \:y = - \dfrac{15}{8}$

Hence,

$\rm \implies\:Point \: of \: intersection \: is \: \bigg( - \dfrac{9}{8}, \: - \dfrac{15}{8} \bigg)$

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Slope or gradient of line segment joining the points (a, b) and (c, d) is represented by m and is given by

$\red{\rm :\longmapsto\:\boxed{\tt{ \: Slope \: of \: line, \: m = \frac{d - b}{c - a} \: }}}$

So,

$\rm :\longmapsto\:Slope \: of \: line \: joining \: (3, - 2) \: and \: \bigg( - \dfrac{9}{8}, \: - \dfrac{15}{8} \bigg) \: is$

$\rm :\longmapsto\:Slope = \dfrac{ - \dfrac{15}{8} + 2}{ - \dfrac{9}{8} - 3}$

$\rm :\longmapsto\:Slope = \dfrac{ \dfrac{ - 15 + 16}{8}}{\dfrac{ - 9 - 24}{8}}$

$\bf\implies \:Slope = - \dfrac{1}{33}$

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Learn More :-

Different forms of equations of a straight line

1. Equations of horizontal and vertical lines

Equation of line parallel to x - axis passes through the point (a, b) is x = a.

Equation of line parallel to x - axis passes through the point (a, b) is x = a.

2. Point-slope form equation of line

Equation of line passing through the point (a, b) having slope m is y - b = m(x - a)

3. Slope-intercept form equation of line

Equation of line which makes an intercept of c units on y axis and having slope m is y = mx + c.

4. Intercept Form of Line

Equation of line which makes an intercept of a and b units on x - axis and y - axis respectively is x/a + y/b = 1.

5. Normal form of Line

Equation of line which is at a distance of p units from the origin and perpendicular makes an angle β with the positive X-axis is x cosβ + y sinβ = p.