Answer:
[tex]y=\frac{5}{9}x+\frac{-83}{9}[/tex]
Step-by-step explanation:
Given equation of line:
[tex]9x+5y=3[/tex]
To find the equation of line perpendicular to the line of the given equation and passes through point (4,-7).
Writing the given equation of line in standard form.
Subtracting both sides by [tex]9x[/tex]
[tex]9x-9x+5y=3-9x[/tex]
[tex]5y=3-9x[/tex]
Dividing both sides by 5.
[tex]\frac{5y}{5}=\frac{3}{5}-\frac{9x}{5}[/tex]
[tex]y=\frac{3}{5}-\frac{9x}{5}[/tex]
Rearranging the equation in standard form [tex]y=mx+b[/tex]
[tex]y=-\frac{9x}{5}+\frac{3}{5}[/tex]
Applying slope relationship between perpendicular lines.
[tex]m_1=-\frac{1}{m_2}[/tex]
where [tex]m_1[/tex] and [tex]m_2[/tex] are slopes of perpendicular lines.
For the given equation in the form [tex]y=mx+b[/tex] the slope [tex]m_2[/tex]can be found by comparing [tex]y=-\frac{9x}{5}+\frac{3}{5}[/tex] with standard form.
∴ [tex]m_2=-\frac{9}{5}[/tex]
Thus slope of line perpendicular to this line [tex]m_1[/tex] would be given as:
[tex]m_1=-\frac{1}{-\frac{9}{5}}[/tex]
∴ [tex]m_1=\frac{5}{9}[/tex]
The line passes through point (4,-7)
Using point slope form:
[tex]y-y_1=m(x-x_1)[/tex]
Where [tex](x_1,y_1)\rightarrow (4,-7)[/tex] and [tex]m=m_1=\frac{5}{9}[/tex]
So,
[tex]y-(-7)=\frac{5}{9}(x-4)[/tex]
Using distribution.
[tex]y+7=(\frac{5}{9}x)-(\frac{5}{9}\times 4)[/tex]
[tex]y+7=\frac{5}{9}x-\frac{20}{9}[/tex]
Subtracting 7 to both sides.
[tex]y+7-7=\frac{5}{9}x-\frac{20}{9}-7[/tex]
Taking LCD to subtract fractions
[tex]y=\frac{5}{9}x-\frac{20}{9}-\frac{63}{9}[/tex]
[tex]y=\frac{5}{9}x+\frac{(-20-63)}{9}[/tex]
[tex]y=\frac{5}{9}x+\frac{-83}{9}[/tex]
[tex]y=\frac{5}{9}x-\frac{83}{9}[/tex]
Thus, the equation of line in standard form is given by:
[tex]y=\frac{5}{9}x-\frac{83}{9}[/tex]
