Question

1. Find the equation of the straight line which passes
through the point (2,-3) and the point of intersect
of the lines x + y + 4 = 0 and 3x - y - 8 = 0.
​

Answer 1

Answer:

Equation of the line = 2x - y - 7 = 0

Step-by-step explanation:

Given:

• The line passes through the point (2, -3)
• The line passes through the point of intersection of the lines x + y + 4 = 0 and 3x - y - 8 = 0

To Find:

• Equation of the line

Solution:

Let us first find the point of intersection of the two lines x + y + 4 = 0 and 3x - y - 8 = 0

By cross multiplication method,

$\tt x = \bigg(\dfrac{b_1c_2-b_2c_1}{a_1b_2-a_2b_1} \bigg), y = \bigg(\dfrac{a_2c_1-a_1c_2}{a_1b_2-a_2b_1} \bigg)$

where a₁ = 1, b₁ = 1, c₁ = 4, a₂ = 3, b₂ = -1, c₂ = -8

Substitute the data,

$\tt (x,y)=\bigg(\dfrac{-8+4}{-1-3} ,\dfrac{12+8}{-1-3} \bigg)$

$\tt (x,y)=\bigg(\dfrac{-4}{-4} ,\dfrac{20}{-4} \bigg)$

$\tt (x,y) = (1,-5)$

Now the equation of the line passing through two points is given by,

$\tt y-y_1=\dfrac{y_2-y_1}{x_2-x_1} (x-x_1)$

where x₁ = 2, x₂ = 1, y₁ = -3, y₂ = -5

Substitute the data,

$\tt y+3=\dfrac{-5+3}{1-2} (x-2)$

$\tt y + 3 = 2(x - 2)$

$\tt y + 3 = 2x-4$

$\tt 2x-y-7=0$

Hence the equation of the line is 2x - y - 7 = 0

Answer 2

Step-by-step explanation:

______________________________

$\text{\Large\underline{\red{Question:-}}}$

:$\Longrightarrow$ ● Find the equation of the straight line which passes through the point (2,-3) and the point of intersect of the lines x + y + 4 = 0 and 3x - y - 8 = 0.

$\text{\Large\underline{\orange{To\ find:-}}}$

$\sf To\ find = \begin{cases} \sf{●\ In\ this\ question\ we\ have\ to\ find\ the\ equation\ of\ line.} \end{cases}$

$\text{\Large\underline{\green{Given:-}}}$

$\sf Given = \begin{cases} \sf{●\ The\ line\ passes\ through\ the\ point\ (2,\ -3)} \\ \\ \sf{●\ The\ line\ passes\ through\ the\ point\ of\ intersection\ of\ the\ lines\ x\ +\ y\ +\ 4\ =\ 0\ and\ 3x\ -\ y\ -\ 8\ =\ 0.} \end{cases}$

$\text{\Large\underline{\purple{Solution:-}}}$

Given lines:-

● $\sf{x\ +\ y\ +\ 4\ =\ 0}$

● $\sf{x\ =\ -y\ -4}$..........(1).

● $\sf{3x\ -\ y\ -8\ =\ 0}$..........(2).

From equation (1) and (2):-

● $\sf{3\ (-y\ -4)\ -\ y\ -\ 8\ =\ 0}$

● $\sf{-3y\ -\ 12\ -\ y\ -\ 8\ =\ 0}$

● $\sf{-4y\ =\ 20}$

● $\sf{y\ =\ -5}$

From equation (1) we get:-

● $\sf{x\ =\ +5\ -\ 4}$

● $\sf{x\ =\ 1.}$

Points of interaction P (1, -5).

● Equation of line from point A (2, -3) and P (1, -5),

● $\sf{y\ +\ 3\ =\ \dfrac{-5+3}{1-2}\ (x\ -\ 2)}$

● $\sf{y\ +\ 3\ =\ \dfrac{-2}{-1}\ (x\ -\ 2)}$

● $\sf{y\ +\ 3\ =\ 2x\ -\ 4}$

● $\sf\pink{2x\ -\ y\ -\ 7\ =\ 0.}$

Hence:-

● The equation of line is 2x - y - 7 = 0.

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