Parallel Lines, Slopes, and X-Intercepts: A Comprehensive Guide

Understanding the properties of parallel lines and how to calculate key points like the x-intercept is fundamental in intermediate algebra and geometry. This guide explores the relationships between these concepts using a series of examples and detailed explanations.

Understanding Parallel Lines and Slopes

Two lines are parallel if and only if their slopes are equal. This means that if one line has a certain slope, any line parallel to it will have the same slope. In mathematical terms, if we have two points on a line, (x_1, y_1) and (x_2, y_2), the slope (m) of this line can be calculated using the formula:

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

Example 1: Finding the Equation of a Line and Its Parallel Line

Let's start by finding the equation of the line passing through points (-1, -3) and (1, 5).

Step 1: Calculate the slope.

The slope (m) is calculated as:

[{m frac{5 - (-3)}{1 - (-1)} frac{8}{2} 4}]

Step 2: Write the equation of the line.

Using the slope-intercept form (y mx b), where (b) is the y-intercept. We can find (b) by substituting one of the points into the equation.

If we use point (1, 5), we have:

[{5 4(1) b implies b 1}]

Step 3: Write the equation of the line.

The equation of the line is:

[{y 4x 1}]

Parallel Line with a Given Point

Now, let's find the equation of a second line that is parallel to the first line and passes through the point (2, 3).

Step 1: Use the same slope.

Since the lines are parallel, the slope is still 4.

The equation is of the form:

[{y 4x c}]

Step 2: Find the y-intercept (c).

Substitute the point (2, 3) into the equation:

[{3 4(2) c implies c 3 - 8 -5}]

Step 3: Write the final equation.

The equation of the line is:

[{y 4x - 5}]

Conclusion and Key Points

To find the x-intercept of a line, set (y 0) in the equation and solve for (x). Here are the key points we've covered:

The slope of parallel lines is the same. The equation of a line in slope-intercept form is (y mx c). To find the x-intercept of a line, set (y 0) and solve for (x).

In the given problem, we found that the x-intercept of the second line is (x frac{5}{4} 1.25).