Problem 62 - Introductory Algebra for College Students

Question

In Exercises \(61-64\), 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 \((2,2)\) and the origin has slope 1

Step-by-Step Solution

Answer

The statement is true - the line through (2,2) and the origin does indeed have a slope of 1.

Step 1: Identify the Given Points

Identify the points given in the exercise; these are (2,2) and the origin (0,0).

Step 2: Apply the Slope Formula

Plug the coordinates of the given points into the slope formula. Let the origin (0,0) be \(x_1\), \(y_1\) and (2,2) be \(x_2\), \(y_2\). Thus, our slope \(m\) becomes \(\frac{2 - 0}{2 - 0} = 1.\)

Step 3: Compare and Conclude

Compare calculated slope with the slope given in the statement. Since the calculated slope is 1 and the slope in the statement is also 1, it can be concluded that the statement is indeed true.'

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. This method combines algebra and Euclidean geometry to describe the position of points, lines, and shapes. Each point on a plane is defined by an ordered pair of numbers known as coordinates. For example, the point (2,2) represents a point that is 2 units away from the origin (0,0) along the x-axis, and 2 units up along the y-axis. Coordinate geometry is vital for solving problems involving distances, midpoints, gradients, and areas.

Slope Formula

The slope of a line is a measure of its steepness and can be calculated using the slope formula, which is defined as the change in the y-coordinates divided by the change in the x-coordinates between two distinct points on the line. Mathematically, it is expressed as \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points. If a line goes upward from left to right, the slope is positive, and if it goes downward, the slope is negative. Horizontal lines have a slope of 0, while vertical lines have an undefined slope.

True or False Statements

In mathematics, true or false statements are used to assess the validity of certain propositions or problems. In the context of coordinate geometry and linear equations, these statements might concern the properties of a line, such as its slope or y-intercept. For instance, a statement that a line has a certain slope can be verified or refuted by applying the slope formula to two points on the line. Similarly, a statement about the relationship between two linear equations can be evaluated for its truthfulness. Understanding how to test these statements is fundamental to problem solving in algebra and geometry.

Linear Equations

Linear equations represent straight-line graphs in coordinate geometry and are fundamental to algebra. They are typically written in the form \( y = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept, or the y-coordinate of the point where the line crosses the y-axis. A linear equation provides a relationship that connects any x-value with a corresponding y-value, which creates a series of points that lie on a straight line when plotted on a graph. These equations are used to model real-world phenomena and to find solutions to various mathematical problems.