Write an equation (a) in standard form and (b) in slope-intercept form for each line described. through (4, 1), parallel to y=-5

Verified step by step guidance

1. Identify the slope of the given line. The line is given as \(y = -5\), which is a horizontal line with slope \(m = 0\).
2. Since the new line is parallel to \(y = -5\), it must also have the same slope \(m = 0\).
3. Use the point-slope form of a line equation with the point \((4, 1)\) and slope \$0\(: \)y - y_1 = m(x - x_1)\(, which becomes \)y - 1 = 0(x - 4)$.
4. Simplify the equation from the point-slope form to get the slope-intercept form \(y = 1\).
5. Rewrite the equation in standard form. Since \(y = 1\) can be written as \$0x + y = 1\(, the standard form is \)0x + y = 1$.

Key Concepts

Equation of a Horizontal Line

A horizontal line has a slope of zero and is represented by an equation of the form y = k, where k is the y-coordinate of any point on the line. Since the line is parallel to y = -5, it must also be horizontal with the same slope.

Slope-Intercept Form

The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. For horizontal lines, the slope m is zero, simplifying the equation to y = b.

Standard Form of a Line

The standard form of a linear equation is Ax + By = C, where A, B, and C are integers, and A ≥ 0. For horizontal lines, this form typically looks like 0·x + 1·y = C, or simply y = C.