For each line described, write an equation in (a) slope-intercept form, if possible, and (b) standard form. through (3, -5), parallel to y = 4

Verified step by step guidance

1. Identify the slope of the given line. The equation is \( y = 4 \), which is a horizontal line. This means the slope \( m = 0 \).
2. Since the new line is parallel to \( y = 4 \), it must also have the same slope \( m = 0 \).
3. Use the point-slope form of a line equation with the point \( (3, -5) \) and slope \( m = 0 \): \( y - y_1 = m(x - x_1) \). Substitute \( y_1 = -5 \), \( x_1 = 3 \), and \( m = 0 \) to get \( y - (-5) = 0(x - 3) \).
4. Simplify the equation to slope-intercept form \( y = mx + b \). Since \( m = 0 \), the equation becomes \( y = -5 \).
5. Convert the slope-intercept form \( y = -5 \) to standard form \( Ax + By = C \). Rearranging gives \( 0x + y = -5 \), or simply \( y = -5 \).

Key Concepts

Slope-Intercept Form of a Line

The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. It clearly shows the slope and where the line crosses the y-axis, making it easy to graph and understand the line's behavior.

Parallel Lines and Their Slopes

Parallel lines have identical slopes. If a line is parallel to y = 4, which is a horizontal line with slope 0, then the new line must also have slope 0, meaning it is horizontal.

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. It is useful for solving systems of equations and provides a different way to represent linear equations compared to slope-intercept form.