Question

use the given conditions to write an equation for the line in point - slope form and in slope - intercept form.
passing through (4, - 2) and perpendicular to the line whose equation is $y = \frac{1}{4}x + 4$

write an equation for the line in point - slope form.
(simplify your answer. use integers or fractions for any numbers in the equation.)
write an equation for the line in slope - intercept form.
(simplify your answer. use integers or fractions for any numbers in the equation.)

Explanation:

Step1: Find the slope of the perpendicular line

The given line is \( y = \frac{1}{4}x + 4 \), so its slope \( m_1=\frac{1}{4} \). For perpendicular lines, the product of slopes is -1, so \( m_1\times m_2=-1 \). Solving for \( m_2 \), we get \( m_2 = -4 \).

Step2: Write the point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(4,-2) \) and \( m = - 4 \). Substituting these values, we have \( y-(-2)=-4(x - 4) \), which simplifies to \( y + 2=-4(x - 4) \).

Step3: Convert to slope - intercept form

Start with \( y + 2=-4(x - 4) \). Distribute the - 4 on the right side: \( y+2=-4x + 16 \). Then subtract 2 from both sides: \( y=-4x+16 - 2 \), so \( y=-4x + 14 \).

Answer:

• Point - slope form: \( y + 2=-4(x - 4) \)
• Slope - intercept form: \( y=-4x + 14 \)