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How do we find a standard form equation for the line with $( - 10, - 6)$ and $(10, - 8)$ ?

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Last updated date: 17th Jun 2024
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Answer
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Hint: To solve this question, first we will find the slope of the given coordinates of the line. And then we will find the slope-intercept. After determining both the slope and y-intercept, now we can conclude the Standard form equation with the help of slope and slope-intercept.

Complete step by step solution:
First, we need to determine the slope of the line. The slope can be found by using the formulae:
$m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
where $m$ is the slope and $({x_1},{y_1})$ and $({x_2},{y_2})$ are the two points on the line.
Substituting the values from the points in the problem gives:
$m = \dfrac{{ - 8 - ( - 6)}}{{10 - ( - 10)}} = \dfrac{{ - 8 + 6}}{{10 + 10}} = \dfrac{{ - 2}}{{20}} = - \dfrac{1}{{10}}$
We can now write and equation in slope-intercept form. The slope-intercept form of a linear equation is:
$y = mx + b$
Where $m$ is the slope and $b$ is the y-intercept value.
Substituting the slope we calculated and the y-intercept we determined gives:
$y = - \dfrac{1}{{10}}x + ( - 6 - ( - 10))$
$\because y = - \dfrac{1}{{10}}x + 4$
We need to convert this equation to the Standard Form for Linear Equations. The Standard form of a linear equation is:
$Ax + By = C$
Where, if at all possible, A, B and C are integers, and A is non-negative, and A,B and C have no common factors other than 1
First, we can add $\dfrac{1}{{10}}x$ to each side of the equation to put the $x$ and $y$ variables on the left side of the equation as required by the Standard Formula:
$
  \dfrac{1}{{10}}x + y = \dfrac{1}{{10}}x + - \dfrac{1}{{10}}x - 4 \\
   \Rightarrow \dfrac{1}{{10}}x + y = 0 - 4 \\
   \Rightarrow \dfrac{1}{{10}}x + y = - 4 \\
 $
We can now multiply each side of the equation by 10 to eliminate the fraction as required by the Standard Formula:
$
  10(\dfrac{1}{{10}}x + y) = 10 \times - 4 \\
   \Rightarrow (10 \times \dfrac{1}{{10}}x) + (10 \times y) = - 40 \\
   \Rightarrow x + 10y = - 40 \\
 $

Hence, the Standard Form equation for the given line is $x + 10y = - 40$.

Note:
The slope and the intercept define the linear relationship between two variables and can be used to estimate an average rate of change. The greater the magnitude of the slope, the steeper the line and the greater the rate of change.