Problem 359 Use slopes and \(y\) -intercepts... [FREE SOLUTION]

Chapter 4: Problem 359

Use slopes and \(y\) -intercepts to determine if the lines are parallel. \(y=-1 ; \quad y=2\)

Short Answer

Expert verified

The lines are parallel because they both have a slope of zero.

Step by step solution

Identify the form of each equation

Notice that both equations, given in the format of horizontal lines, are in the form of constant values of y: For the first line: \( y = -1 \) For the second line: \( y = 2 \)

Determine the slopes

For horizontal lines, the slope is always zero. Therefore, both lines have a slope of: \( m = 0 \)

Compare the slopes

Check if the slopes are the same: Since both lines have a slope of zero \( m = 0 \), they are parallel.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope

In the context of parallel lines, slope is a key concept. The slope of a line measures its steepness and is usually denoted by the letter \( m \). The slope is calculated as the ratio of the change in the y-coordinate to the change in the x-coordinate between two points on the line. This can be algebraically represented as \( m = \frac{\Delta y}{\Delta x} \). For parallel lines, their slopes are always equal. Hence, if you have two lines and their slopes are identical, they are parallel.

y-intercept

The y-intercept is another important concept that helps in understanding the properties of a line. The y-intercept is the point where the line crosses the y-axis. It is usually denoted by the letter \( b \) in the slope-intercept form of a line equation, which is \( y = mx + b \). Here, \( b \) gives the y-coordinate of the point on the graph where the line meets the y-axis. It's important to know that even though two lines may be parallel, their y-intercepts can be different, which is why they don鈥檛 coincide and instead run parallel without ever intersecting.

horizontal lines

Horizontal lines are a special case of straight lines in geometry. These lines run from left to right and are parallel to the x-axis. In mathematical terms, the equation of a horizontal line is simple: \( y = c \), where \( c \) is a constant. Therefore, the y-coordinate of every point on such a line is the same, which means the slope \( m \) of a horizontal line is zero. In this exercise, both lines \( y = -1 \) and \( y = 2 \) are horizontal, hence they have the same slope of zero and are parallel. Horizontal lines never intersect if their y-values are different.

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91影视

Use slopes and \(y\) -intercepts to determine if the lines are parallel. \(y=-1 ; \quad y=2\)

Short Answer

Step by step solution

Identify the form of each equation

Determine the slopes

Compare the slopes

Key Concepts

slope

y-intercept

horizontal lines

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