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

Identify the given information: the line passes through the point (4, 1) and is parallel to the line given by the equation $y = -5$.

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 $k$.

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 $y = 1$.

Write the equation in standard form. For a horizontal line, the standard form is $y = 1$, which can also be written as $0x + y = 1$.

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 $y = 1$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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