Question

find the equation of the line that is perpendicular to $y = -\frac{1}{7}x + 7$ and contains the point (2,16).
y = ?x +

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the given line \( y = -\frac{1}{7}x + 7 \) is \( m_1 = -\frac{1}{7} \). For two perpendicular lines, the product of their slopes is -1, i.e., \( m_1 \times m_2 = -1 \). Let the slope of the perpendicular line be \( m_2 \). So, \( -\frac{1}{7} \times m_2 = -1 \). Solving for \( m_2 \), we get \( m_2 = 7 \) (by multiplying both sides by -7).

Step2: Use point - slope form to find the equation

The point - slope form of a line is \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(2,16) \) and \( m = 7 \). Substituting these values, we have \( y - 16 = 7(x - 2) \).

Step3: Simplify the equation to slope - intercept form

Expand the right - hand side: \( y - 16 = 7x-14 \). Then, add 16 to both sides of the equation: \( y=7x - 14 + 16 \), which simplifies to \( y = 7x+2 \).

Answer:

The equation of the line is \( y = 7x + 2 \), so the first box is \( 7 \) and the second box is \( 2 \).