what is the equation of the line that is perpendicular to and has the same y - intercept as the given…

Explanation:

Step1: Find the slope of the given line

The given line passes through points $(0,1)$ and $(5,0)$. The slope $m_1$ of a line passing through $(x_1,y_1)$ and $(x_2,y_2)$ is $m_1=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0 - 1}{5 - 0}=-\frac{1}{5}$.

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 perpendicular line be $m_2$. Then $m_1\times m_2=-1$. Substituting $m_1 = -\frac{1}{5}$, we get $-\frac{1}{5}\times m_2=-1$, so $m_2 = 5$.

Step3: Determine the y - intercept

The y - intercept of the given line is the y - coordinate of the point where the line crosses the y - axis. The given line crosses the y - axis at the point $(0,1)$, so the y - intercept $b = 1$.

Step4: Write the equation of the line

The slope - intercept form of a line is $y=mx + b$. Substituting $m = 5$ and $b = 1$, we get $y = 5x+1$.

Answer:

C. $y = 5x + 1$