what is an equation of the line that passes through the point (5, 4) and is perpendicular to the line 5x +…

Explanation:

Step1: Find the slope of the given line

Rewrite $5x + 4y=4$ in slope - intercept form $y = mx + b$ (where $m$ is the slope). $4y=-5x + 4$, so $y=-\frac{5}{4}x + 1$. The slope of this line is $m_1=-\frac{5}{4}$.

Step2: Find the slope of the perpendicular line

If two lines are perpendicular, the product of their slopes is $- 1$. Let the slope of the required line be $m_2$. Then $m_1\times m_2=-1$. $-\frac{5}{4}\times m_2=-1$, so $m_2=\frac{4}{5}$.

Step3: Use the point - slope form to find the equation of the line

The point - slope form is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(5,4)$ and $m = \frac{4}{5}$. $y - 4=\frac{4}{5}(x - 5)$. Expand the right - hand side: $y - 4=\frac{4}{5}x-4$. Add 4 to both sides to get the slope - intercept form: $y=\frac{4}{5}x$.

Answer:

$y=\frac{4}{5}x$