Problem 26 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the distinct lines through each pair of points are parallel. $$(-7,6)\( and \)(0,4) ;(-9,-3)\( and \)(1,5)$$

Step-by-Step Solution

Answer

The lines are not parallel because their slopes are not equal.

Step 1: Calculate Slope of First Line

The slope (m1) of a line through two points (x1, y1) and (x2, y2) can be found using the formula \( m = (y2-y1) / (x2-x1) \). Plugging in the coordinates of the first pair of points (-7,6) and (0,4), the slope of the first line (m1) can be calculated as \( m1 = (4-6) / (0 - (-7)) = -2/7 \).

Step 2: Calculate Slope of Second Line

Next, calculate the slope of the second line (m2) using the same formula and the coordinates of the second pair of points (-9,-3) and (1,5). This gives us \( m2 = (5 - (-3)) / (1 - (-9)) = 8/10 = 4/5 \).

Step 3: Compare Slopes of the Two Lines

After calculating the slopes of the two lines, compare them. If the slopes are equal, then the two lines are parallel. Here, it can be seen that m1 is not equal to m2 (-2/7 ≠ 4/5), which means the lines are not parallel.

Key Concepts

Slope Calculation

The concept of slope is fundamental in understanding how lines behave on a coordinate plane. The slope of a line represents how steep the line is and is crucial in determining the direction and angle of the line. To calculate the slope of a line given two points, we use the formula:

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

This formula calculates the change in the vertical direction \((y_2 - y_1)\) divided by the change in the horizontal direction \((x_2 - x_1)\).

For example, to find the slope of a line passing through the points \((-7,6)\) and \((0,4)\), substitute these values into the formula:

• \( m_1 = \frac{4 - 6}{0 - (-7)} = \frac{-2}{7} \)

Similarly, for the points \((-9,-3)\) and \((1,5)\):

• \( m_2 = \frac{5 - (-3)}{1 - (-9)} = \frac{8}{10} = \frac{4}{5} \)

Calculating slope is a foundational step in analyzing linear equations and relationships in coordinate geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to represent geometric shapes and solve problems. By using the coordinate plane, we can explore relationships between points, lines, and figures through algebraic expressions.

Each point on the plane is defined by a pair of coordinates \((x, y)\). Lines are drawn through these points, and various properties like slope and intercepts help us describe these lines accurately.

• Point Example: \((-7,6)\) is a specific location on the plane.
• Line Representation: Two points define a line whose characteristics can be analyzed using algebraic methods.

Coordinate geometry allows us to perform operations such as finding the distance between points, determining midpoints, and analyzing parallel or perpendicular lines. The connection between algebra and geometry in this field is what makes it such a powerful tool in mathematics.

Comparing Slopes

When it comes to parallel lines in coordinate geometry, comparing slopes is a straightforward method to determine their relationship. Lines are parallel if and only if they have the same slope.

In our problem, we calculated the slopes:

• Slope of the first line, \(m_1 = \frac{-2}{7}\)
• Slope of the second line, \(m_2 = \frac{4}{5}\)

These values are not equal, indicating that the lines through these points are not parallel.

This comparison highlights a crucial aspect of parallel lines: they never intersect, and their identical slopes are a direct reflection of this property. By understanding how to compare slopes, we gain insight into the geometric arrangement and alignment of lines on the plane.