Answer :
Answer:
[tex]y = \displaystyle \frac{5}{2}\, x + 54[/tex].
(Equivalently, [tex]5\, x - 2\, y = 108[/tex].)
Step-by-step explanation:
Rewrite the equation of the line [tex]2\, x + 5\, y = 10[/tex] in the slope-intercept form [tex]y = m\, x + b[/tex] to find the slope [tex]m[/tex].
[tex]5\, y = -2\, x + 10[/tex].
[tex]\displaystyle y = -\frac{2}{5}\, x + 2[/tex].
Therefore, the slope of the line [tex]2\, x + 5\, y = 10[/tex] is [tex](-2 / 5)[/tex].
Let [tex]m_1[/tex] and [tex]m_2[/tex] denote the slope of two lines on a cartesian plane. If these two lines are perpendicular to one another, then [tex]m_1 \cdot m_2 = (-1)[/tex].
In this question, the slope of the first line was already found [tex]m_1 = (-2 / 5)[/tex]. Solve [tex]m_1 \cdot m_2 = (-1)[/tex] for the slope [tex]m_2[/tex] of the unknown line:
[tex]\begin{aligned}m_2 &= \frac{-1}{m_1} = \frac{-1}{-2 / 5} = \frac{5}{2}\end{aligned}[/tex].
The following information are thus available for the requested line:
- The slope of this line is [tex]\displaystyle m = \frac{5}{2}[/tex].
- This line contains the point [tex](x_0,\, y_0) = (-20,\, 4)[/tex].
Hence, the point-slope form [tex](y - y_0) = m\, (x - x_0)[/tex] would be a possible choice for the equation of this line:
[tex]\displaystyle y - 4 = \frac{5}{2}\, (x - (-20))[/tex].
Simplify to obtain the slope-intercept form equation of this line:
[tex]\displaystyle y = \frac{5}{2}\, x + 54[/tex].
Equivalently:
[tex]5\, x - 2\, y = 108[/tex].