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Determine whether two lines are parallel using their slopes

Parallel lines in the coordinate plane have the same slope, because their rate of change and direction are identical; this can be determined from graphs, pairs of points, or linear equations. The reasoning includes recognizing horizontal lines as having slope zero and distinct vertical lines as parallel despite undefined slope, while distinguishing coincident lines and avoiding the misconception that perpendicular slopes indicate parallelism.

Detailed Explanation: Determine whether two lines are parallel using their slopes

Lines are parallel when they have the same slope and are not the same line. The slope tells how much a line rises or falls as it moves to the right.

Use the slope formula:

m=y2−y1x2−x1m=\frac{y_2-y_1}{x_2-x_1}

Example: Determine whether the line through (1,2)(1,2) and (4,8)(4,8) is parallel to the line through (−2,1)(-2,1) and (1,7)(1,7).

Step 1: Find the slope of the first line.

m1=8−24−1=63=2m_1=\frac{8-2}{4-1}=\frac{6}{3}=2

Step 2: Find the slope of the second line.

m2=7−11−(−2)=63=2m_2=\frac{7-1}{1-(-2)}=\frac{6}{3}=2

Step 3: Compare the slopes.

Both slopes are 22. The lines have the same direction, so they are parallel, as long as they are not the same line. Since the points are different and the lines do not overlap, the lines are parallel.

Conclusion:

The two lines are parallel.\boxed{\text{The two lines are parallel.}}

Remember:

  • Horizontal lines have slope 00. Distinct horizontal lines are parallel.
  • Distinct vertical lines are also parallel, even though their slopes are undefined.
  • Perpendicular lines have slopes that are different, so they are not parallel.

Learn by doing: Determine whether two lines are parallel using their slopes

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