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Write the equation of the line perpendicular to 2x+3y=9 that passes through (-2,5). Write your answer in slope-intercept form. Show your work.

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ANSWER

[tex]y = \frac{3}{2} x + 8[/tex]



EXPLANATION

The line given to us has equation,

[tex]2x + 3y = 9[/tex]

We need to write this equation in the slope intercept form to obtain,


[tex]3y = - 2x + 9[/tex]



[tex]\Rightarrow \: y = - \frac{2}{3}x + 3 [/tex]


The slope of this line is

[tex]m_1 = - \frac{2}{3} [/tex]
Let the slope of the perpendicular line be

[tex]m_2[/tex]

Then
[tex]m_1 \times m_2 = - 1[/tex]


[tex] - \frac{2}{3} m_2= - 1[/tex]

This implies that,

[tex]m_2 = - 1 \times - \frac{3}{2} [/tex]


[tex]m_2 = \frac{3}{2} [/tex]



Let the equation of the perpendicular line be,

[tex]y = mx + b[/tex]

We substitute the slope to get,


[tex]y = \frac{3}{2} x + b[/tex]

Since this line passes through
[tex](-2,5)[/tex]
it must satisfy its equation.


This means that,

[tex]5= \frac{3}{2} ( - 2)+ b[/tex]


[tex]5 = - 3 + b[/tex]



[tex]5 + 3 = b[/tex]


[tex]b = 8[/tex]

Wherefore the slope-intercept form is

[tex]y = \frac{3}{2} x + 8[/tex]

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