Question

Find the equation of a line passing through the point (-5, 2) parallel to the line 7x – 3y = 12.​

Answer 1

EXPLANATION.

Equation of a line passing through the point (-5,2) parallel to the line,

⇒ 7x - 3y = 12.

Slope of a parallel line = -a/b.

Slope of line 7x - 3y = 12 = -7/-3 = 7/3.

Co-ordinate = (-5,2).

Equation of line.

⇒ (y - y₁) = m ( x - x₁).

⇒ ( y - 2 ) = 7/3( x -(-5)).

⇒ 3 ( y - 2 ) = 7 ( x + 5 ).

⇒ 3y - 6 = 7x + 35.

⇒ 3y - 7x = 41.

                                               

MORE INFORMATION.

Monotonic function.

These are of two types.

(1) = Monotonic Increasing.

If the value of f(x) should increase (decrease) or remain equal by increasing (decreasing) the value of x.

x₁ < x₂ ⇒ f(x₁) ≤ f(x₂).

x₁ < x₂ ⇒ f(x₁) ≠ f(x₂).

where ∨ x₁, x₂ ∈ D.

x₁ > x₂ ⇒ f(x₁) ≥ f(x₂)

x₁ > x₂ ⇒ f(x₁) ≠ f(x₂)

where ∨ x₁, x₂ ∈ D.

(2) = Monotonic Decreasing.

If the value of f(x) should decrease ( increase) or remain equal by increasing ( decreasing ) the value of x.

x₁ < x₂ ⇒  f(x₁) ≥ f(x₂).

x₁ < x₂ ⇒ f(x₁) ≠ f(x₂).

where x₁, x₂ D.

x₁ > x₂ ⇒ f(x₁) ≤ f(x₂).

x₁ > x₂ ⇒  f(x₁) ≠ f(x₂).

where x₁, x₂ D.

(3) = A function is said to be monotonic function in a domain if it is either monotonic increasing or monotonic decreasing in that domain.

(4) = If x₁ < x₂ ⇒ f(x₁) < f(x₂), ∨ x₁, x₂ ∈ D, then f(x) is called strictly increasing in domain D.

(5) = If x₁ < x₂ ⇒ f(x₁) > f(x₂), ∨ x₁, x₂ ∈ D, then it is called strictly decreasing in domain D.

Answer 2

$\underline\blue{\bold{Given \: Question :- }}$

Find the equation of a line passing through the point (-5, 2) parallel to the line 7x – 3y = 12.

__________________________________________

$\huge \orange{AηsωeR} ✍$

${ \boxed {\bf{Given}}}$

• A line passing through the point (-5, 2) parallel to the line 7x – 3y = 12.

${ \boxed {\bf{To \: Find}}}$

• Equation of line

${ \boxed {\bf{Formula \: used :- }}}$

If the equation of line is ax + by + c = 0, then slope is

$\bf \:m = - \dfrac{coefficient \: of \: x}{coefficient \: of \: y} = - \dfrac{a}{b}$

Condition for parallel lines :-

Two lines having slope m and n are parallel if and only if

$\bf \:  ⟼ m \: = n$

Slope - point form :-

Let us consider a line which passing through the point (a, b) having slope m, then equation of line is given by

$\bf \:y - b = m( - a)$

___________________________________________

${ \boxed {\bf{Solution}}}$

⟼ Let the required line be L, which passing through the point (-5, 2) parallel to the line 7x – 3y = 12.

⟼ Let slope of line L be 'm'.

Now equation of given line is

$\bf \:  ⟼ 7x - 3y = 12 \: ⟼ \: (1)$

$\bf \:  ⟼ So, \: slope \: of \: (1) =\dfrac{ - 7}{ - 3} = \dfrac{7}{3}$

$\bf \:  ⟼ Since, \: line \: L \: is \: parallel \: to \: line \: (1)$

$\bf\implies \: \: Slope \: of \: line \: L = slope \: of \: (1)$

$\bf\implies \:m = \dfrac{7}{3}$

⟼ So, equation of line L, passing through (-5, 2) and having slope, m = 7/3 is given by

$\bf \:  ⟼ y - 2 = \dfrac{7}{3} (x - ( - 5))$

$\bf \:  ⟼ 3y - 6 = 7x + 35$

$\bf \:  ⟼ 7x - 3y + 41 = 0$

⟼ So, required equation of line L is 7x - 3y + 41 = 0.

_____________________________________

$\large \red{\bf \: ⟼ Explore \: more } ✍$

What is a Straight Line?

• A straight line is defined as a line traced by a point travelling in a constant direction with zero curvature. In other words, the shortest distance between two points is called a straight line.

 

Slope of a Line

• Tan θ is called the slope or gradient of line l if θ is the inclination of a line l. The slope of a line whose inclination is not equal to 900. It is denoted by ‘m’.

Slope – Point Form

• Assume that P0(a, b) is a fixed point on a non-vertical line L, whose slope is m. Consider that P (x, y) be an arbitrary point on L then the point (x, y) lies on the line with slope m through the fixed point (a, b), if and only if, its coordinates satisfy the following equation y – b = m (x – a)

Two – Point Form

• The equation of the line passing through the given points (x1, y1) and (x2, y2) is defined by

$\bf \:y-y_{1}=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}(x-x_{1})$

Slope-Intercept Form

• Assume that a line L with slope m cuts the y-axis at a distance c from the origin where the distance c is called the y-intercept of the line L. Therefore, the coordinates of the point where the line meets the y-axis are (0, c). So, the line L has slope m and passes through a fixed point (0, c). Thus, from the slope – point form, the equation of the line L is y – c =m( x -0 )
• Therefore, the point (x, y) on the line with slope m and y-intercept c lies on the line if and only if y = m x +c

Intercept Form

• Consider a line L that makes x-intercept and y-intercept b on the axes. So that, L meets x-axis at the point (a, 0) and y-axis at the point (0, b). Thus, the equation of the line having the intercepts a and b on x-and y-axis respectively is given by

$\bf \:\dfrac{x}{a}+\dfrac{y}{b}=1$

________________________________________