For each line described, write an equation in (a) slope-intercept form, if possible, and (b) standard form. through (-7, 4), perpendicular to y = 8

Verified step by step guidance

1. Identify the given line's equation: \( y = 8 \). This is a horizontal line where the slope \( m = 0 \).
2. Since the new line is perpendicular to \( y = 8 \), its slope will be the negative reciprocal of 0. The negative reciprocal of 0 is undefined, which means the new line is vertical.
3. A vertical line passing through the point \((-7, 4)\) has the equation \( x = -7 \). This cannot be written in slope-intercept form \( y = mx + b \) because the slope is undefined.
4. For the standard form of a vertical line, rewrite \( x = -7 \) as \( x + 0y = -7 \). This fits the standard form \( Ax + By = C \) where \( A = 1 \), \( B = 0 \), and \( C = -7 \).
5. Summarize: (a) Slope-intercept form is not possible for this vertical line, and (b) the standard form is \( x = -7 \) or equivalently \( x + 0y = -7 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a line is y = mx + b, where m represents the slope and b is the y-intercept. This form makes it easy to identify the slope and where the line crosses the y-axis, which is essential for graphing and understanding linear relationships.

Perpendicular Lines and Their Slopes

Two lines are perpendicular if the product of their slopes is -1. For a horizontal line like y = 8 (slope 0), the perpendicular line must be vertical, which has an undefined slope. Recognizing this helps determine the correct slope for the new line.

Standard Form of a Line

The standard form of a line is Ax + By = C, where A, B, and C are integers, and A ≥ 0. This form is useful for solving systems of equations and provides a clear, consistent way to represent lines, especially vertical or horizontal ones.