Finding the Equation of a Line Parallel to Another Through a Given Point

In this article, we will walk through the process of finding the equation of line L2, which is parallel to line L1 and passes through a specific point. This topic is fundamental in understanding the relationship between parallel lines and their equations. Let's break it down step by step.

Given Information

Line L1 passes through the points (-6, 1) and (-2, -1). Line L2 is parallel to L1 and passes through the point (5, 7). Our goal is to find the equation of L2.

Step 1: Calculate the Slope of Line L1

The slope of a line passing through two points ((x_1, y_1)) and ((x_2, y_2)) is given by the formula:

[ m frac{y_2 - y_1}{x_2 - x_1} ]

Substituting the given points: (-6, 1) and (-2, -1), we get:

[ m_1 frac{-1 - 1}{-2 - (-6)} frac{-2}{4} -frac{1}{2} ]

Therefore, the slope of line L1 is (-frac{1}{2}).

Step 2: Confirm the Slope for Line L2

Since line L2 is parallel to line L1, they will have the same slope. Thus, the slope of line L2, (m_2), is also (-frac{1}{2}).

Step 3: Use the Point-Slope Form to Find the Equation of L2

The point-slope form of a line is given by:

[ y - y_1 m(x - x_1) ]

Substituting the slope (m -frac{1}{2}) and the point (5, 7) into the formula, we get:

[ y - 7 -frac{1}{2}(x - 5) ]

Now, we simplify the equation:

[ y - 7 -frac{1}{2}x frac{5}{2} ]

Adding 7 to both sides:

[ y -frac{1}{2}x frac{5}{2} 7 ]

Converting 7 to a fraction with a denominator of 2:

[ 7 frac{14}{2} ]

Therefore:

[ y -frac{1}{2}x frac{5}{2} frac{14}{2} -frac{1}{2}x frac{19}{2} ]

Final Equation of L2

The equation of line L2 in slope-intercept form is:

[ y -frac{1}{2}x 9.5 ]

This can also be written in standard form, but the slope-intercept form is the most useful for understanding the relationship between the slope and the y-intercept.

In summary, the equation of line L2 is:

[ y -frac{1}{2}x 9.5 ]

Additional Considerations

Understanding the relationship between parallel lines and their equations is crucial in many areas of geometry and algebra. This method can be applied to solve more complex problems involving parallel lines and their equations. Pay attention to the slope and how it relates to the given points to find the desired equation.

For further exploration, you might want to look into the point-slope form, the general form of a line, and the relationship between the slopes of parallel and perpendicular lines.

Keywords: slope, parallel lines, equation of a line