Understanding Parallel Lines and Their Equations

When dealing with lines in geometry and algebra, a fundamental concept that often comes up is that of parallel lines. Parallel lines have a unique property: they never intersect and maintain a constant distance between them. This is often related to the slope of the lines, which we will explore in this article. Specifically, we will learn how to find the equation of a line that is parallel to a given line and passes through a specified point.

Given Line: 2x - y - 7 0

Let's start by finding the equation of a line parallel to the given line (2x - y - 7 0) and passing through the point ((3, -2)).

Step 1: Determine the Slope of the Given Line

The first step in finding the equation of a parallel line is to determine the slope of the given line. We need to rewrite the equation in the slope-intercept form, (y mx b), where (m) is the slope and (b) is the y-intercept.

Starting with the given equation:

[2x - y - 7 0]

Isolate (y):

[-y -2x 7]

Multiplying by -1:

[y 2x - 7]

From this, we can see that the slope (m) of the given line is 2.

Step 2: Utilize the Slope of the Parallel Line

Since parallel lines have the same slope, the slope of the line that is parallel to the given line is also 2.

Step 3: Use the Point-Slope Form of the Line Equation

Now that we know the slope, we can use the point-slope form of the line equation to find the equation of the line that passes through the point ((3, -2)).

The point-slope form is:

[y - y_1 m(x - x_1)]

Here, (x_1 3) and (y_1 -2). Substituting these values into the equation:

[y - (-2) 2(x - 3)]

Simplify:

[y 2 2x - 6]

Moving terms to isolate (y):

[y 2x - 8]

Thus, the equation of the line that is parallel to (2x - y - 7 0) and passes through the point ((3, -2)) is (textbf{y 2x - 8}).

Alternative Methods to Find the Equation

There are multiple ways to find the equation of a parallel line, and each method reveals a different aspect of the line's properties.

Method 1: Utilizing Coefficients

The coefficients of (x) and (y) in the equation of a parallel line are the same. Therefore, we need to find the value of (q) such that the equation (2x - y - q 0) passes through the point ((3, -2)).

Substituting the point into the equation:

[2(3) - (-2) - q 0]

[6 2 - q 0]

[8 - q 0]

[q 8]

This gives us the equation:

[2x - y - 8 0]

Method 2: Using the Point and Slope

We can also derive the equation using the slope and the point directly. The slope (m 2) because it is the same for the parallel line. Using the point-slope form, we get:

[y - (-2) 2(x - 3)]

[y 2 2x - 6]

[y 2x - 8]

Again, the equation simplifies to:

[y 2x - 8]

Method 3: Using the Slope Directly

Alternatively, we can directly find the constant by using the point ((3, -2)) and the slope (m 2).

Solving for (b) in the equation (2x - y - 2(3) - (-2) 0):

[2x - y - 6 2 0]

[2x - y - 4 0]

This simplifies to:

[y 2x - 8]

Thus, the equation of the line is:

[y 2x - 8]

Conclusion

In summary, we learned different methods to find the equation of a line that is parallel to a given line and passes through a specified point. These methods involve understanding the slope, point-slope form, coefficients, and direct substitution. Mastering these concepts will help in solving similar problems in geometry and algebra.