Find the equation of the line that is parallel to $2 x + 5 y - 7 = 0$ and passes through the mid-point of the line segment joining the points $(2, 7)$ and $(- 4, 1)$ .

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Solution

Step1: Calculation of the slope of line.
The slope-intercept form of the equation of a line is given by the formula $y = m x + c$ , where $m$ is the slope and $c$ is the $Y$ -intercept of the line.
Simplify the equation $2 x + 5 y - 7 = 0$ to isolate variable $y$ .
$2 x + 5 y - 7 = 0 \Rightarrow 5 y = - 2 x + 7 (s u b t r a c t i n g 2 x - 7 f r o m b o t h s i d e s) ∴ y = - \frac{2}{5} x + \frac{7}{5} (d i v i d i n g b o t h s i d e s b y 5) - e q u a t i o n (1)$
Comparing equation (1) with equation $y = m x + c$ we get the slope of line $2 x + 5 y - 7 = 0$ as $- \frac{2}{5}$ .
The slope of the line that is parallel to $2 x + 5 y - 7 = 0$ is also $- \frac{2}{5}$ .
Step2: Calculation of mid-point of the line segment joining the points $(2, 7)$ and $(- 4, 1)$ .
The co-ordinates of mid-point of a line joining the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ are given by the formula $(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$ .
For points $(2, 7)$ and $(- 4, 1)$ , $x_{1} = 2, y_{1} = 7, x_{2} = - 4, y_{2} = 1$ .
Thus, co-ordinates of mid-point of a line joining the points $(2, 7)$ and $(- 4, 1)$ are given by:
$x = \frac{x_{1} + x_{2}}{2} \Rightarrow x = \frac{2 - 4}{2} \Rightarrow x = \frac{- 2}{2} ∴ x = - 1 y = \frac{7 + 1}{2} \Rightarrow y = \frac{8}{2} ∴ y = 4$
Thus, the mid-point of line joining the points $(2, 7)$ and $(- 4, 1)$ is $(- 1, 4)$ .
Step3: Calculation of equation of the line.
The point-slope form of the equation of a line passing through point $(x_{1}, y_{1})$ and having slope $m$ is given by the formula $(y - y_{1}) = m (x - x_{1})$ .
For a line passing through the point $(- 1, 4)$ and having slope $- \frac{2}{5}$ ; $x_{1} = - 1, y_{1} = 4$ and $m = - \frac{2}{5}$ .
$(y - 4) = - \frac{2}{5} (x - (- 1)) \Rightarrow 5 (y - 4) = - 2 (x + 1) (m u l t i p l y i n g b o t h s i d e s b y 5) \Rightarrow 5 y - 20 = - 2 x - 2 \Rightarrow 5 y - 20 + 2 x + 2 = 0 (a d d i n g 2 x + 2 t o b o t h s i d e s) ∴ 2 x + 5 y - 18 = 0$
Hence, the equation of the line that is parallel to $2 x + 5 y - 7 = 0$ and passes through the mid-point of the line segment joining the points $(2, 7)$ and $(- 4, 1)$ is $2 x + 5 y - 18 = 0$ .

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Find the equation of the line that is parallel to 2x+5y-7=0 and passes through the mid-point of the line segment joining the points (2,7) and (-4,1).

Find the equation of the line that is parallel to $2 x + 5 y - 7 = 0$ and passes through the mid-point of the line segment joining the points $(2, 7)$ and $(- 4, 1)$ .