Problem 64 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The line through \((3,1)\) and \((3,-5)\) has zero slope.

Step-by-Step Solution

Answer

The statement is false. The correct statement should be: The line through points \((3,1)\) and \((3,-5)\) has an undefined slope.

Step 1: Remembering Slope's Definition

The slope \( m \) of any straight line on a coordinate plane is defined as the ratio of the vertical to the horizontal distance between any two points \((x_1,y_1)\) and \((x_2,y_2)\) on the line. It is given by the formula \( m = \frac{y2−y1}{x2−x1} \). However, when \( x2 = x1 \), the expression results in a division by zero, which is undefined in mathematics.

Step 2: Applying the Slope Formula to the Given Points

Trying to find the slope with the given points \((3,1)\) and \((3,-5)\) we substitute \(x_1 = 3\), \(y_1 = 1\), \(x_2 = 3\), \(y_2 = -5\) into the formula: \( m = \frac{-5-1}{3-3} \). Calculating the numerator, we get -6. However, the denominator equals 0, which makes the entire fraction undefined. So, the slope for the line is not zero but undefined.

Step 3: Correcting the False Statement

Since the slope of the line is not zero but undefined, we recognize that the original statement is false. To correct it, we state: The line through points \((3,1)\) and \((3,-5)\) has an undefined slope.

Key Concepts

undefined slope

Whenever we deal with lines on a coordinate plane, understanding the concept of slope is essential. The slope indicates the steepness and direction of a line. However, some lines do not have a slope we can describe with a numeric value. Instead, we say these lines have an undefined slope. This occurs when we have a vertical line where the x-coordinates of any two points are identical. The slope equation, given by
\( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \),
requires a change in the x-coordinates to calculate the ratio. If there's no change, the denominator is zero, resulting in an undefined expression in mathematics. Thus, a line like the one passing through (3,1) and (3,-5) on a coordinate plane is vertical, and no matter how we try to calculate the slope, we will always end up dividing by zero. This particular quality of vertical lines is a fundamental concept when studying linear relationships in coordinate plane mathematics.

slope formula

One of the critical tools in coordinate plane mathematics is the slope formula, which allows us to determine the rise over run—or in more formal terms, the ratio of the vertical change to the horizontal change between two distinct points on a line. To express this numerically, we use the formula
\( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \),
where \(m\) represents the slope, and \((x_1, y_1)\) and \((x_2, y_2)\) represent two points on the line. If we apply this formula to two points (3,1) and (3,-5), noting that the x-coordinates do not change, we encounter a division by zero scenario, which indicates an undefined slope. In practice, the proper application of this slope formula is essential for graphing linear equations, comparing the steepness of lines, and solving for unknowns within a coordinate system. Understanding how to manipulate and apply the formula correctly is crucial for mastering problems involving linear equations.

coordinate plane mathematics

The study of coordinate plane mathematics involves numerous concepts, where the plotting of points and understanding the relationship between those points is fundamental. The plane is divided into four quadrants, with each point represented by an ordered pair (x, y). Lines on the plane can suggest numerous mathematical relationships and properties. For example, the slope of a line tells us how it inclines or declines as it moves from left to right. The points (3,1) and (3,-5) lie on a vertical line, showcasing an instance where standard slope calculations do not apply, resulting in an undefined slope.

In understanding the coordinate plane, it is also essential to learn about different types of slopes—positive, negative, zero, and undefined. Each type gives us a visual understanding of how lines behave. Zero slope lines are horizontal, positive slope lines rise to the right, negative slope lines fall to the right, and lines with undefined slopes are vertical. These principles allow students to visualize functions and solve geometric problems involving lines and, ultimately, to find success in further mathematical concepts such as calculus or analytic geometry.