Problem 24 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the distinct lines through each pair of points are parallel. $$(2,4)\( and \)(6,1) ;(-3,1)\( and \)(1,-2)$$

Step-by-Step Solution

Answer

Yes, the distinct lines through each pair of points are parallel because they have the same slope of -3/4.

Step 1: Determine the Slope of the First Line

The equation to calculate the slope of a line between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m_1 = (y_2 - y_1) / (x_2 - x_1)\). For the first pair of points, \((2,4)\) and \((6,1)\), we substitute into the equation and get \(m_1 = (1-4) / (6-2) = -3/4\).

Step 2: Determine the Slope of the Second Line

Similarly, the slope of the second line passing through the points \((-3,1)\) and \((1,-2)\) is calculated by \(m_2 = (y_2 - y_1) / (x_2 - x_1)\). Substituting the points into this equation, we get \(m_2 = (-2-1) / (1 - (-3)) = -3/4\).

Step 3: Compare the Slopes

Next, the slopes of the two lines are compared. If the slopes are equal, then the lines are parallel. From step 1 and 2, we found that \(m_1 = m_2 = -3/4\). Therefore, the lines are indeed parallel. Note: When calculating the slopes, ensure that the calculation of slope does not result in division by zero, which would signify a vertical line.

Key Concepts

Slope Calculation

The slope of a line is a measure of its steepness and direction. It is calculated by the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Here, \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of two distinct points on the line.
To compute the slope, subtract the y-coordinate of the first point from the y-coordinate of the second point, then do the same with their x-coordinates and divide the results.
In our example, for the points \( (2,4) \) and \( (6,1) \), the slope is:

• \( m_1 = \frac{1 - 4}{6 - 2} = \frac{-3}{4} \)

Calculating the slope helps determine how much the line rises or falls as you move from one point to the next. A positive slope indicates the line rises, while a negative slope indicates it falls. It's crucial for analyzing and comparing linear relationships in mathematics, such as identifying parallel lines.

Equation of a Line

The equation of a line is essential for describing linear relationships in coordinate geometry. The simplest form of this equation is known as the slope-intercept form, represented as:

• \( y = mx + b \)

Where \( m \) is the slope, and \( b \) is the y-intercept. This formula conveys that the value of \( y \) can be determined from any \( x \) value if the slope and y-intercept are known.
Once you have the slope, as calculated, and a point through which the line passes, you can substitute these values into the equation to solve for \( b \), the y-intercept.
Understanding the equation allows us to visualize the line on a coordinate plane where every solution is a point on the line. In the case of parallel lines, the slope remains unchanged, signifying lines that never meet or diverge as they both extend indefinitely.

Coordinate Geometry

Coordinate geometry, also referred to as analytic geometry, bridges algebra and geometry by using a coordinate system. It empowers the analysis of geometric shapes via algebra, using an organized grid of horizontal and vertical lines known as axes.
Points in coordinate geometry are plotted as \( (x, y) \) pairs representing locations in this grid. By calculating distances and gradients (slopes) between these points, coordinate geometry aids in solving complex geometric problems algebraically.

• Helps quantify geometric relationships by assigning numerical values to geometric figures.
• Allows for precise plotting and visual representation of lines, curves, and shapes.

In our exercise, coordinate geometry provides a systematic method to assess whether lines are parallel, by comparing their slopes. It is a powerful tool that simplifies the study of shapes, angles, and dimensions in two-dimensional space.