Write an equation (a) in standard form and (b) in slope-intercept form for each line described. through (4, -4), perpendicular to x=4

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Identify the given line: The line is described as \( x = 4 \), which is a vertical line passing through all points where \( x = 4 \).

Determine the slope of the given line: Since \( x = 4 \) is vertical, its slope is undefined. A line perpendicular to a vertical line must be horizontal, which means its slope is \( 0 \).

Use the point-slope form to write the equation of the line passing through the point \( (4, -4) \) with slope \( 0 \). The point-slope form is \( y - y_1 = m(x - x_1) \), so substitute \( m = 0 \), \( x_1 = 4 \), and \( y_1 = -4 \) to get \( y - (-4) = 0(x - 4) \).

Simplify the equation from step 3 to get the slope-intercept form \( y = b \). Since the slope is zero, the equation simplifies to \( y = -4 \).

Write the equation in standard form: For a horizontal line \( y = -4 \), the standard form is \( 0x + y = -4 \), or simply \( y = -4 \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Equation of a Vertical Line

A vertical line has an undefined slope and is represented by an equation of the form x = a constant. For example, x = 4 is a vertical line passing through all points where x equals 4.

Perpendicular Lines and Their Slopes

Two lines are perpendicular if the product of their slopes is -1. Since a vertical line has an undefined slope, its perpendicular line is horizontal with a slope of 0.

Forms of Linear Equations: Standard and Slope-Intercept

The standard form of a line is Ax + By = C, where A, B, and C are constants. The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Both forms express the same line differently.