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How do you find the equation in slope intercept form, of the line passing through of the line passing through the points $( - 1,2)$ and $(3, - 4)$ ?

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Last updated date: 17th Jun 2024
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Answer
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Hint: In this question, we are given two points and we have to find the equation of the line passing through these two points. The standard slope-intercept equation of a line is $y = mx + c$ where m is the slope of this line and c is the x-intercept of the line. From the 2 given points, we will find the slope of the line and then plug in the values in the slope-intercept form. Or we can directly use the equation $y - {y_1} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}(x - {x_1})$ to find out the equation of the line passing through the points $( - 1,2)$ and $(3, - 4)$ .

Complete step by step solution:
The given line is passing through the points $( - 1,2)$ and $(3, - 4)$
On plugging the known values in the equation $y - {y_1} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}(x - {x_1})$ , we get –
$
  y - 2 = \dfrac{{ - 4 - 2}}{{3 - ( - 1)}}[x - ( - 1)] \\
   \Rightarrow y - 2 = \dfrac{{ - 6}}{4}(x + 1) \\
   \Rightarrow y - 2 = \dfrac{{ - 3}}{2}(x + 1) \\
   \Rightarrow 2(y - 2) = - 3(x + 1) \\
   \Rightarrow 2y - 2 = - 3x - 3 \\
   \Rightarrow 2y + 3x + 1 = 0 \\
 $
Hence, the equation of the line passing through the points $( - 1,2)$ and $(3, - 4)$ is $2y + 3x + 1 = 0$

Note: In mathematics, slope or also known as the gradient of a line is a number that describes both the direction and the steepness of the line. It is often denoted by the letter “m”. For a line present in a plane containing x and y axes, the slope of the line is given as the ratio of the change in the y-coordinate and the corresponding change in the x-coordinate between two distinct points of the line, that is, the slope of a line joining two points $({x_1},{y_1})$ and $({x_2},{y_2})$ is given by the formula $\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$ . We can solve the above question by first calculating the slope by using this formula.