Find the Function Perpendicular to y=-1/5x+3 Through Point (2,9)

To solve this problem, follow these steps:

Step 1: Identify the slope of the given line.

The given line is $y = -\frac{1}{5}x + 3$. Here, the slope ($m$) is $-\frac{1}{5}$.

Step 2: Determine the slope of the perpendicular line.

For lines to be perpendicular, the product of their slopes must equal $-1$. Hence, if $m_1 = -\frac{1}{5}$ is the slope of the given line, the slope ($m_2$) of the line perpendicular to it can be found using:

$m_1 \times m_2 = -1$, which implies:

$-\frac{1}{5} \times m_2 = -1$.

Solve for $m_2$ to get:

$m_2 = 5$.

Step 3: Use the point-slope form to find the equation of the line.

The line passes through point $(2, 9)$, and we have determined $m_2 = 5$.

Using the point-slope form $y - y_1 = m(x - x_1)$, substitute $m = 5$, $x_1 = 2$, $y_1 = 9$:

$y - 9 = 5(x - 2)$.

Step 4: Simplify the equation.

Distribute the 5:

$y - 9 = 5x - 10$.

Add 9 to both sides to solve for $y$:

$y = 5x - 1$.

Therefore, the equation of the line we are looking for is $\boxed{y = 5x - 1}$.