To find an equation of a line that passes through the origin and is perpendicular to the line given by the equation [tex]\(5x - 2y = 8\)[/tex], follow these steps:
1. Determine the slope of the given line:
The given line’s equation is [tex]\(5x - 2y = 8\)[/tex].
First, we'll rewrite this equation in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
[tex]\[
5x - 2y = 8
\][/tex]
Solving for [tex]\(y\)[/tex], we get:
[tex]\[
-2y = -5x + 8
\][/tex]
[tex]\[
y = \frac{5}{2}x - 4
\][/tex]
Therefore, the slope ([tex]\(m_1\)[/tex]) of the line [tex]\(5x - 2y = 8\)[/tex] is [tex]\(\frac{5}{2}\)[/tex].
2. Find the slope of the perpendicular line:
For a line to be perpendicular to another, its slope ([tex]\(m_2\)[/tex]) should be the negative reciprocal of the slope of the original line.
The negative reciprocal of [tex]\(\frac{5}{2}\)[/tex] is:
[tex]\[
m_2 = -\frac{1}{\frac{5}{2}} = -\frac{2}{5}
\][/tex]
3. Write the equation of the perpendicular line:
The line we are looking for passes through the origin, which means it has a y-intercept [tex]\(b = 0\)[/tex] when written in slope-intercept form [tex]\(y = mx + b\)[/tex].
With the slope [tex]\(m_2 = -\frac{2}{5}\)[/tex] and [tex]\(b = 0\)[/tex], the equation of the line becomes:
[tex]\[
y = -\frac{2}{5}x
\][/tex]
4. Convert the equation to standard form:
To write this equation in standard form [tex]\(Ax + By = C\)[/tex], we rearrange it:
[tex]\[
y = -\frac{2}{5}x
\][/tex]
Multiply both sides by 5 to eliminate the fraction:
[tex]\[
5y = -2x
\][/tex]
Rearrange to get:
[tex]\[
2x + 5y = 0
\][/tex]
Thus, the equation of the line that passes through the origin and is perpendicular to the line [tex]\(5x - 2y = 8\)[/tex] is:
[tex]\[
2x + 5y = 0
\][/tex]
