Question

How do you find the slope and $y-$ intercept of the line $2x-3y=18?$

Answer 1

Here as you can see that we have to find the slope $y-$ intercept of the line $2x-3y=18$
For finding the slope and y- intercept of line use the standard form of linear equation.
Here is the standard form of any linear equation i.e. $y=mx+b$
Where, $m$ is the slope of the equation and $b$ is the $y$-intercept.
The slope of the equation of the form.
$ax+by=c$ is
Find Slope of line of the given equation by using following formula:
$m=\dfrac{-a}{b}$
And for finding $y-$intercept of the given equation put $x=0$ in the equation.

Complete step by step solution:
As you know that given, equation is in the form of $ax+by=c$
i.e. $2x-3y=18$
Where, $a=2$
$\Rightarrow b=-3$
$\Rightarrow$ $c=18$
To find the slope of the equation slope of the line $m=\dfrac{-a}{b}$
$\Rightarrow$ $=\dfrac{-2}{-3}$
Here, $\left( - \right)$ minus in numerator and denominator gets canceled.
Slope of line $m=\dfrac{2}{3}$
To find the $y$-intercept of the equation $2x-3y=18$ put $x=0$
$\Rightarrow$ $2x-3y=18$
$\Rightarrow$ $2\left( 0 \right)-3y=18$
Any number multiplied by $'0'$ will be $0$
$\Rightarrow$ $0-3y=18$
$\Rightarrow$ $-3y=18$
Here $'-3'$ will be transferred to the right side for finding $y$-intercept.
$\Rightarrow$ $y=\dfrac{-18}{3}$
As $'18'$ comes $'6'$ times in $3$table, therefore the value of $y$-intercept will be,
$\Rightarrow$ $y=\dfrac{6}{1}$
$\Rightarrow$ $y=-6$

**Hence, slope of line is $\dfrac{2}{3}$ and $y$-intercept of the line is $-6$ or $\left( 0,-6 \right)$
**

Additional Information:
Any linear equation has the form of $y=mx+b$
$m$ is the slope of the equation and $b$ is the $y$-intercept.
The slope of the line $m$ is found by
$\Rightarrow$ $m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
Where $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ are the coordinates of any two points in the line.
The $y$-intercept $b,$found by plugging in $x=0$ into the equation which results in $y=b$ and therefore is the $y$-intercept. In some cases if the equation is already. Arranged for you nicely, like$y=3x+5$. We can easily find the $y$-intercept for this line, which is $5.$
Other lines the equation might not be arranged nicely, with cases such as $\dfrac{1}{2}x+3y=5$
In which we solve the $y$-intercept.
$\Rightarrow$ $\dfrac{1}{2}x+3y=4$
$\Rightarrow$ $3y=\dfrac{4-1}{2}x$
$\Rightarrow$ $y=\dfrac{-\dfrac{1}{2}x+4}{3}$
$\Rightarrow$ $y=-\dfrac{1}{6}x+\dfrac{4}{3}$
So, comparing above equation by $y=mx+b$
Therefore, slope of line $m=-\dfrac{1}{6}$ and $y$-intercept of line $b=\dfrac{4}{3}$

Note: Compare the equation with the standard equation $ax+by=c$
Identity $a,b$ and $c.$
Formula for finding slope of line $m=\dfrac{-a}{b}$ and for finding $y$-intercept put $x=0$ in the given equation.
Use the different basics of arithmetic for finding the value of slope of line and $y$-intercept i.e. division, addition, subtraction, multiplication etc.