Two lines are graphed below. What can we conclude about them? Select all that apply. coordinate plane showing y equals 3 x plus 1 and y equals negative one third x minus 2 The lines are perpendicular. The lines are parallel. The lines have the same slope. The lines have opposite reciprocal slopes.
Mathematics · Middle School · Thu Feb 04 2021
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From the given information, the two lines are graphed and their equations are provided as y = 3x + 1 and y = -1/3x - 2. To understand the relationship between the two lines, we should look at the slopes of the lines.
The slope of the first line, y = 3x + 1, is 3. The slope of the second line, y = -1/3x - 2, is -1/3.
- Perpendicular lines are lines that intersect at a right angle (90 degrees) and have slopes that are opposite reciprocals of each other. Since 3 and -1/3 are indeed opposite reciprocals (because 3 * -1/3 = -1), we can conclude the lines are perpendicular. - Parallel lines are lines in the same plane that never intersect and have the same slope. Since the slopes of these two lines are not the same, they are not parallel.
Therefore, the conclusions we can draw are: - The lines are perpendicular. - The lines have opposite reciprocal slopes.
The other options are not correct: - The lines are not parallel. - The lines do not have the same slope.
Extra: When analyzing lines on a graph to determine their relationship to each other, there are key concepts to keep in mind:
- Slope is a measure of how steep a line is and is usually denoted by the letter 'm'. In the equation of a line in the form y = mx + b, m represents the slope, while b represents the y-intercept.
- Parallel lines have identical slopes, meaning that the values of 'm' in their equations are the same. However, the y-intercepts (b values) may differ.
- Perpendicular lines have slopes that are negative reciprocals of each other. This means if one line has a slope of 'a', the other line that is perpendicular to it will have a slope of '-1/a'.
- If two lines have different slopes, they will eventually intersect at some point (unless they are parallel), and the angle at which they intersect can tell us if they are perpendicular or not.
Understanding these concepts can help students analyze the relationships between lines on a graph and their equations.