Introduction
 Linear equations are equations of degree one. For example, $3x5y=17$ is a linear equation in two variables.
 The equation of a straight line is also a linear equation.
Slope or Gradient of a line
 The slope of a line is the ratio of change in the ycoordinates to the change in the xcoordinates.
 The slope $\left(m\right)$ of a nonvertical line passing through the points $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ is given by: $m=\frac{{y}_{2}{y}_{1}}{{x}_{2}{x}_{1}},$${x}_{1}\ne {x}_{2}$.
 The formula for slope is also referred to as rise over run.
 The slope of a line parallel to the xaxis is zero.
 The slope of the yaxis is not defined.
 If two lines are parallel, their slopes are equal.
 If two lines are perpendicular to each other, the product of their slopes is 1.
Forms of Linear Equation
 The standard form of a line:

 In the standard form, a linear equation is expressed as: $Ax+By=C$, where $A,B\ne 0$.
 The SlopeIntercept form of a line:

 The equation of a line with slope $m$, and making an intercept $c$ on yaxis is given by $y=mx+c$.
 The Point Slope form of a line:

 The equation of a line with slope $m$, and passing through a point $\left({x}_{1},{y}_{1}\right)$ is given by $y{y}_{1}=m\left(x{x}_{1}\right)$.
 The Two Point form of a line:

 The equation of a line passing through the points $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ is $\left(y{y}_{1}\right)=\frac{{y}_{2}{y}_{1}}{{x}_{2}{x}_{1}}\left(x{x}_{1}\right)$.
 Equation of a line in Intercept form:

 The equation of a line that makes intercepts of lengths $a$ and $b$ on the xaxis and yaxis respectively is given by $\frac{x}{a}+\frac{y}{b}=1$.
Solved Examples
Question 1: Find the slope of a line passing through the points (3, 5) and (7, 4).
Solution: Here, ${x}_{1}=3,{y}_{1}=5$, ${x}_{2}=7,\mathrm{and}{y}_{2}=4$
Slope, $m=\frac{{y}_{2}{y}_{1}}{{x}_{2}{x}_{1}}$$=\frac{45}{7\left(3\right)}$$=\frac{1}{7+3}$$=\frac{1}{10}$
Question 2: Find the equation of a line with slope, $m=\frac{2}{5}$ and passing through the point $\left(2,1\right)$.
Solution: The pointslope form of a line is given by the equation $\left(y{y}_{1}\right)=m\left(x{x}_{1}\right)$
Here, ${x}_{1}=2,{y}_{1}=1$, and $m=\frac{2}{5}$.
So, the desired equation will be $\left(y1\right)=\frac{2}{5}\left[x\left(2\right)\right]$
$5\left(y1\right)=2\left(x+2\right)$
$5y5=2x+4$
$2x5y+9=0$
Question 3: Find the equation of a line passing through the points (1, 3) and (2, 5).
Solution: Here, ${x}_{1}=1,{y}_{1}=3,{x}_{2}=2,and{y}_{2}=5$.
slope $m=\frac{\left({y}_{2}{y}_{1}\right)}{\left({x}_{2}{x}_{1}\right)}$$=\frac{5\left(3\right)}{21}$$=\frac{5+3}{1}$$=8$
The equation of the line will be $\left(y{y}_{1}\right)=m\left(x{x}_{1}\right)$
$\left[y\left(3\right)\right]=8\left(x1\right)$
$\left(y+3\right)=8x8$
$y+3=8x8$
$8xy=11$
Cheat Sheet
 Linear equations are equations in which the highest power (degree) of the variable is 1.
 The slope $\left(m\right)$ of a nonvertical line passing through the points $\left({x}_{1},{y}_{1}\right)$, and $\left({x}_{2},{y}_{2}\right)$ is given by $m=\frac{{y}_{2}{y}_{1}}{{x}_{2}{x}_{1}},$${x}_{1}\ne {x}_{2}$.
 Equation of a line in standard form: $Ax+By=C$, where $A$ and $B$ are nonzero coefficients of x and y, respectively.
 The SlopeIntercept form of a line: The equation of a line with slope $m$, and making an intercept $c$ on yaxis is given by $y=mx+c$.
 The Point Slope form of a line: The equation of a line with slope $m$, and passing through a point $\left({x}_{1},{y}_{1}\right)$ is given by $y{y}_{1}=m\left(x{x}_{1}\right)$.
 The Two Point form of a line: The equation of a line passing through the points $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$ is $y{y}_{1}=\frac{{y}_{2}{y}_{1}}{{x}_{2}{x}_{1}}\left(x{x}_{1}\right)$.
 Equation of a line in Intercept form: The equation of a line that makes intercepts of lengths $a$ and $b$ on the xaxis and yaxis respectively is given by $\frac{x}{a}+\frac{y}{b}=1$.
 Two lines are parallel if and only if their slopes are equal.
 Two lines are perpendicular if and only if the product of their slopes is 1.
Blunder Areas
 Remember that the slope of a vertical line is not defined.
 In the slopeintercept form of a line, $y=mx+c$, $c$ is the yintercept and must not be confused with the xintercept.
 Abhishek Tiwari
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