Answer:
[tex]D = \frac{25\sqrt{13}}{13}[/tex]
Step-by-step explanation:
Given
[tex]y = 4 - \frac{2}{3}x[/tex]
[tex](x_1,y_1) = (-2,-3)[/tex]
Required
Determine the distance
[tex]y = 4 - \frac{2}{3}x[/tex]
Write the above equation in standard form:
[tex]Ax + By + C = 0[/tex]
So, we have:
[tex]\frac{2}{3}x+y - 4 = 0[/tex]
By comparison:
[tex]A = \frac{2}{3}[/tex] [tex]B = 1[/tex] and [tex]C = -4[/tex]
The distance is calculated using:
[tex]D = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}[/tex]
Where:
[tex](x_1,y_1) = (-2,-3)[/tex]
[tex]A = \frac{2}{3}[/tex] [tex]B = 1[/tex] and [tex]C = -4[/tex]
This gives:
[tex]D = \frac{|\frac{2}{3} * (-2) + 1 * (-3) - 4|}{\sqrt{(\frac{2}{3})^2 + 1^2}}[/tex]
[tex]D = \frac{|-\frac{4}{3} -3 - 4|}{\sqrt{\frac{4}{9} + 1}}[/tex]
Take LCM
[tex]D = \frac{|\frac{-4-9-12}{3}|}{\sqrt{\frac{4+9}{9}}}[/tex]
[tex]D = \frac{|\frac{-25}{3}|}{\sqrt{\frac{13}{9}}}[/tex]
[tex]D = |\frac{-25}{3}|/\sqrt{\frac{13}{9}}[/tex]
[tex]D = \frac{25}{3}/\sqrt{\frac{13}{9}}[/tex]
Split the square root
[tex]D = \frac{25}{3}/\frac{\sqrt{13}}{\sqrt{9}}[/tex]
Change / to *
[tex]D = \frac{25}{3}*\frac{\sqrt{9}}{\sqrt{13}}[/tex]
[tex]D = \frac{25}{3}*\frac{3}{\sqrt{13}}[/tex]
[tex]D = \frac{25}{\sqrt{13}}[/tex]
Rationalize
[tex]D = \frac{25}{\sqrt{13}} * \frac{\sqrt{13}}{\sqrt{13}}[/tex]
[tex]D = \frac{25\sqrt{13}}{13}[/tex]
Hence, the distance is:
[tex]D = \frac{25\sqrt{13}}{13}[/tex]
