Question

How do you find the slope and y-intercept of $2x-5y=10$?

Answer 1

We are given an equation line as $2x-5y=10$. We are asked to find the slope of it, we will first learn what is slope then we focus on various methods to find the slope. We learn to find the slope using $\tan \theta$ and also by $m=\dfrac{\text{rise}}{\text{run}}$. We will learn about slope intercept form; we convert our problem to slope intercept form then we will find the slope. We then finally learn about y-intercept and use $n=0$ for the equation to find the y-intercept.

Complete step-by-step solution:
We are given an equation of a as $2x-5y=10$ we are asked to find the slope of the given equation and also to find the y-intercept. We will first understand that what is the slope means then we will focus on the ways to find the slope of any given equation.
Now the slope of any line is the angle made by the line with the positive x-axis.
We generally find the slope by finding the ratio of rise and run.
Rise means movement of the function along the y-axis while run refers to the movement along the x-axis.
So one way in slope $m=\dfrac{\text{rise}}{\text{run}}$
Other way is to find tan of the angle made by the line with x-axis so slope= $\tan \theta$

Slope is eliminated as m, so
$m=\tan \theta$, or $m=\dfrac{\text{rise}}{\text{run}}$
Other ways to find the slope is to use the equation given to us.
General equation of line in standard form is given as: $ax+by+c=0$
So, we can convert this equation to slope intercept form given as $y=mx+c$
When $m$ is slope, c is to the y-interpret. So, we can find a slope from here. Now we have $2x-5y=10$. So, we will transform them into slope intercept for. So we will use diagram tools to change it into the slope intercept and then we will get the slope and y-intercept.
We have
$2x-5y=10$
So we subtract $2x$ on both side, we get
$2x-5y-2x=10-2x$
Simplifying we get
$-5y=-2x+10$
Now divide both sides by $-5$, we get
$\begin{aligned} & y=\dfrac{-2x+10}{-5} \\\ &\Rightarrow y={\dfrac{2}{5}}x + (-2) \\\ &\Rightarrow y=\dfrac{2}{5}x - 2 \\\ \end{aligned}$
Comparing above equation with slope intercept form $y=mx+c$
We get $m=\dfrac{2}{5}\,\,\,\text{and}\,\,\,c=-2$
So item slope is $m= \dfrac{2}{5}$ and y-intercept is $-2$

Note: Another way which is too difficult to find slope in as we know standard form of line in given as $ax+by+c=0$ then we know slope is given as $\dfrac{-a}{b}$ where a is coefficient and x and b is coefficient of y.
Now in our equation $2x-5y=10$ we have
$\begin{aligned} & a=2 \\\ & b=-5 \\\ \end{aligned}$
So using above trick slope will be
$m=\dfrac{-a}{b}$
Using above value we get
$m=\dfrac{-2}{-5}$
Simplifying we get,
$m=\dfrac{2}{5}$
And the other way to find y-intercept is to put $x=0$ in our equation and solve for y. In $2x-5y=10$ we put $x=0$, We will get
$-5y=10$
Divide both side by $-5$, we get
$y=-2$
So y-intercept is $-2$