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 given information: the line passes through the point (4, 1) and is parallel to the line given by the equation \(y = -5\).
2. Recall that the equation \(y = -5\) represents a horizontal line where the y-coordinate is always -5, so any line parallel to it is also horizontal and has the form \(y = k\) for some constant \(k\).
3. Since the new line passes through the point (4, 1), substitute the y-coordinate of this point into the general form of a horizontal line to find \(k\). This gives \(y = 1\).
4. Write the equation in standard form. For a horizontal line, the standard form is \(y = 1\), which can also be written as \(0x + y = 1\).
5. Write the equation in slope-intercept form, which is \(y = mx + b\). Since the line is horizontal, the slope \(m = 0\), so the equation is \(y = 0x + 1\), simplifying to \(y = 1\).

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 constant y-value for all points on the line. For example, y = -5 is a horizontal line passing through all points with y-coordinate -5.

Parallel Lines

Parallel lines have the same slope but different y-intercepts. Since the given line y = -5 is horizontal with slope 0, any line parallel to it must also have slope 0, meaning it is also horizontal.

Forms of Linear Equations

The standard form of a line is Ax + By = C, where A, B, and C are constants, and the slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Converting between these forms helps express the line's equation clearly.