Question

How do you find the slope and y intercept of $2x-y=1$? \[\]

Answer 1

Hint: We recall the three forms of writing a linear equation which are the general form$Ax+By+C=0$, the slope intercept form $y=mx+c$ and the standard form$Ax+By=C$. We take the term with which $x$ is multiplied to the right hand side of the standard equation $Ax+By=C$ and then divide both by coefficient of $y$ to convert it into slope-intercept form. We use obtained $m,c$ in terms of $A,B,C$ to get the slope point from of $2x-y=1$.\[\]

Complete step by step answer:
We know from the Cartesian coordinate system that every linear equation $Ax+By+C=0$can be represented as a line. If the line is inclined with positive $x-$axis at an angle $\theta $ then its slope is given by $m=\tan \theta $ and if it cuts $y-$axis at a point $\left( 0,c \right)$ from the origin the $y-$intercept is given by $c$. The slope-intercept form of equation is given by
\[y=mx+c....\left( 1 \right)\]
We know that the standard form of linear equation otherwise also known as intercept form is written with constant $C$on the right side of equality sign as
\[Ax+By=C...\left( 2 \right)\]
 Let us subtract $Ax$ from both sides of the above equation to have;
\[By=-Ax+C\]
We divided both side of above equation by $B$ to have
\[y=\dfrac{-A}{B}x+\dfrac{C}{B}.....\left( 3 \right)\]
We see that the above equation is in the slope-intercept form. We compare equation (1) and (3) to have
\[m=\dfrac{-A}{B},c=\dfrac{C}{B}\]
We are given the equation $2x-y=1$ which is in standard form. We compare it with $Ax+By=C$ to have $A=2,B=-1,C=1$. So the required slope $m$ and the required $y-$intercept $c$of the given line are
\[\begin{align}
  & m=\dfrac{-A}{B}=-\dfrac{2}{-1}=2 \\
 & c=\dfrac{C}{B}=\dfrac{1}{-1}=-1 \\
\end{align}\]

Note: We note that the equation of the given line in slope-intercept form is $y=2x-1$. We also note that the standard form is $Ax+By=C$ is also called intercept form because we get $x-$intercept by putting $y=0$ as $\dfrac{-C}{A}$ and similarly $y-$intercept as $\dfrac{-C}{B}$. We note that the slope of the equation gives orientation and inclination of the line with positive $x-$axis. Since here $m=2$ slope is positive the line will be increasing from left to right.