Question

How do you find the slope and intercept of $5x - 2y = 8$ ?

Answer 1

Hint: Firstly, we convert the given line equation into slope-intercept form of a line i.e. $y = mx + c$ and by comparing the values with the terms in the original equation, we can find the slope. Equate the values of $x$ and $y$ to zero to find the intercepts of the line.

Complete step-by-step solution:
The given line equation is $5x - 2y = 8$
Convert this equation into slope-intercept form of a line i.e. $y = mx + c$
$ \Rightarrow 5x - 2y = 8$
Subtract $5x$ from both sides of the equation
$ \Rightarrow 5x - 2y - 5x = 8 - 5x$
$ \Rightarrow - 2y = 8 - 5x$
Divide $ - 2$ from both sides of the equation
$ \Rightarrow \dfrac{{ - 2y}}{{ - 2}} = \dfrac{8}{{ - 2}} - \dfrac{{5x}}{{ - 2}}$
$ \Rightarrow y = - 4 + \dfrac{{5x}}{2}$
Simplify the above equation so that the equation looks like a slope-intercept form of a line
\[ \Rightarrow y = \dfrac{5}{2}x - 4\]
Compare the above equation with $y = mx + c$ to find the value of $m$
Slope $(m) = \dfrac{5}{2}$
$\therefore $ The slope of the equation $5x - 2y = 8$ is $\dfrac{5}{2}$
 Equate the value of $x$ to zero to find the y-intercept of the given equation
$5x - 2y = 8$
$ \Rightarrow 5(0) - 2y = 8$
On further simplification,
$ \Rightarrow 2y = - 8$
$ \Rightarrow y = - 4$
$\therefore $The y-intercept of the equation $5x - 2y = 8$ is $ - 4$
Equate the value of $y$ to zero to find the x-Intercept of the given equation
$5x - 2y = 8$
$ \Rightarrow 5x - 2(0) = 8$
On further simplification,
$ \Rightarrow 5x = 8$
$ \Rightarrow x = \dfrac{8}{5}$

$\therefore $The x-intercept of the equation $5x - 2y = 8$ is $\dfrac{8}{5}$

Additional Information: Slope intercept form: Any equation of the form $y = mx + c$ is called the slope-intercept form. In the equation, $m$ represents the slope of the line equation and $c$ represents the vertical intercept or y-intercept of the line and it is also the value of $y$ when $x = 0$

Note: There is an alternative method to find the intercepts of a line equation. Convert the given line equation into intercept form of a line i.e. $\dfrac{x}{a} + \dfrac{y}{b} = 1$ , where a is the x-intercept and b is the y-intercept
$5x - 2y = 8$
$ \Rightarrow \dfrac{{5x}}{8} - \dfrac{{2y}}{8} = 1$
$ \Rightarrow \dfrac{x}{{\dfrac{8}{5}}} - \dfrac{y}{4} = 1$
By comparing the above equation with intercept form, we get
$\therefore $ X-Intercept = $\dfrac{8}{5}$ , Y-Intercept = $ - 4$