Question

1. the line passes through (4, -4) and is perpendicular to y = -4x - 2. write the equation in standard form.

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the given line \( y = -4x - 2 \) is \( m_1=-4 \). For two perpendicular lines, the product of their slopes is \(-1\), so let the slope of the required line be \( m_2 \). Then \( m_1\times m_2=-1 \), so \( -4\times m_2=-1 \), which gives \( m_2 = \frac{1}{4} \).

Step2: Use point - slope form to find the equation of the line

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(4,-4) \) and \( m = \frac{1}{4} \). Substituting these values, we get \( y - (-4)=\frac{1}{4}(x - 4) \), which simplifies to \( y + 4=\frac{1}{4}x - 1 \).

Step3: Convert to standard form (\(Ax + By = C\))

Subtract \( \frac{1}{4}x \) from both sides: \( -\frac{1}{4}x+y=-1 - 4 \), then multiply through by \(-4\) to get rid of the fraction: \( x-4y = 20 \).

Answer:

The equation of the line in standard form is \(x - 4y=20\)