Answer :

jacob193

Answer:

[tex]\displaystyle y = \left(-\frac{5}{2}\right)\, x + \frac{25}{2}[/tex].

Step-by-step explanation:

The slope-intercept equation of a line [tex]y = m\, x + b[/tex] specifies:

  • the slope [tex]m[/tex] of the line, and
  • the [tex]y[/tex]-intercept [tex]b[/tex] of the line.

To find the line in this question:

  • Start by finding the slope of the line [tex]-2\, x + 5\, y = 10[/tex], which is perpendicular to this line.
  • Find the slope of the line in question using the fact in a cartesian plane, if two lines are perpendicular, and the slope of one of the lines is [tex]m[/tex] ([tex]m \ne 0[/tex]), the slope of the other line would be [tex](-1 / m)[/tex].
  • After finding the slope of the line in question, make use of the fact that the point [tex](3,\, 5)[/tex] is on this line to find the [tex]y[/tex]-intercept.

To find the slope of the line [tex]-2\, x + 5\, y = 10[/tex], rewrite the equation of this line in the slope-intercept form. Make sure that [tex]x[/tex] and [tex]y[/tex] are on different sides of the equation, and the coefficient of [tex]y\![/tex] is [tex]1[/tex].

[tex]5\, y = 2\, x + 10[/tex].

[tex]\displaystyle y = \frac{2}{5}\, x + 2[/tex].

The coefficient of [tex]x[/tex] would be the slope of this line: [tex](2/5)[/tex]. The slope of the line perpendicular to this line would be:

[tex]\displaystyle -\frac{1}{(2/5)} = -\frac{5}{2}[/tex].

Hence, the slope-intercept equation of this line would be [tex]y = (-5/2)\, x + b[/tex] for some constant [tex]b[/tex]. Find the value of [tex]b[/tex] using the fact that the point [tex](3,\, 5)[/tex] is on this line:

[tex]\displaystyle 5 = \left(-\frac{5}{2}\right)\, (3) + b[/tex].

[tex]b = \displaystyle \frac{25}{2}[/tex].

Hence, the slope-intercept equation of the line in question would be:

[tex]\displaystyle y = -\frac{5}{2}\, x + \frac{25}{2}[/tex].