Problem 36 - Elementary Algebra

Question

Sketch the line determined by each pair of points and decide whether the slope of the line is positive, negative, or zero. $$(7,3),(4,-6)$$

Step-by-Step Solution

Answer

The slope is positive (3) and the line rises from left to right.

Step 1: Determine the Slope Formula

The formula to determine the slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 2: Identify the Coordinates

We'll identify our coordinates from the given points: \((x_1, y_1) = (7, 3)\) and \((x_2, y_2) = (4, -6)\).

Step 3: Substitute the Coordinates into the Slope Formula

Substitute the identified coordinates into the slope formula: \[ m = \frac{-6 - 3}{4 - 7} = \frac{-9}{-3} \]

Step 4: Simplify the Slope Expression

Simplifying \( \frac{-9}{-3} \), we get \( m = 3 \). This indicates the slope of the line is positive because 3 is greater than zero.

Step 5: Interpret the Slope Value

A positive slope means that as you move from left to right along the line, the line rises. Thus, the slope is positive for the line through the points \((7,3)\) and \((4,-6)\).

Key Concepts

Slope of a Line

The slope of a line is a crucial concept in algebra, providing insight into how steep a line is on a graph. The slope, often denoted by the letter \(m\), is calculated using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula helps you determine the change in the \(y\)-value relative to the change in the \(x\)-value between two distinct points:

• \((x_1, y_1)\) is the first point on the line.
• \((x_2, y_2)\) is the second point on the line.

A detailed understanding of slope involves recognizing:

• Positive Slope: When the line rises from left to right. For example, the line through \((7,3)\) and \((4,-6)\) has a positive slope because it increases as you move along the line.
• Negative Slope: When the line falls from left to right.
• Zero Slope: Represents a horizontal line, indicating no vertical change as \(x\) changes.
• Undefined Slope: Typically seen in vertical lines, where there is no horizontal change.

Understanding slope is essential for predicting how a line behaves, making it a fundamental tool in grappling with linear equations.

Coordinate Geometry

Coordinate geometry, or analytic geometry, connects algebra and geometry through graphs and coordinates. It allows problems involving shapes, sizes, and relative positions to be solved algebraically by associating each point to pairs of coordinates. Here are the main concepts:

• Points and Coordinates: Each point on a plane is identified by a pair of coordinates, \( (x, y) \), representing its horizontal and vertical positions respectively.
• Distance Formula: Used to find the distance between two points, \((x_1, y_1)\) and \((x_2, y_2)\):\[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \]
• Midpoint Formula: Calculates halfway between two points:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

The importance of coordinate geometry is reflected in its wide array of applications. It provides a way to visualize algebraic relationships and solve geometric problems using algebra. When determining the slope of a line, such as between \((7,3)\) and \((4,-6)\), coordinate geometry plays a vital role by translating these coordinates into a graphical form.

Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate plane to show the relationship that the equation represents. Each linear equation corresponds to a straight line, which is the graphical representation of the equation. Here’s how you approach graphing these equations:

• Start by identifying the equation of the line, typically in the slope-intercept form: \( y = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept, the point where the line crosses the y-axis.
• Calculate the y-intercept: This gives an initial point to plot on the graph.
• Use the slope to determine other points: From the y-intercept, use the slope to find additional points using the rise over run method.
• Plot the points on a graph and draw the line through them.

Graphing gives a visual perspective of relationships described by equations. For instance, by graphing points \((7,3)\) and \((4,-6)\) and drawing the resulting line, one can immediately see the positive slope reflecting the equation's upward trend. Graphing is instrumental in simplifying complex relationships, making it an invaluable tool for understanding linear equations.