Problem 17 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. $$ (4,9),(11,9) $$

Step-by-Step Solution

Answer

The slope is 0, indicating a horizontal line.

Step 1: Understand the Slope Formula

The slope of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \(m\) is the slope.

Step 2: Identify the Points

We are given the points \((4, 9)\) and \((11, 9)\). Assign the first point as \((x_1, y_1) = (4, 9)\) and the second point as \((x_2, y_2) = (11, 9)\).

Step 3: Plug Values into the Slope Formula

Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), substitute \( y_1 = 9 \), \( y_2 = 9 \), \( x_1 = 4 \), and \( x_2 = 11 \). This gives us: \[ m = \frac{9 - 9}{11 - 4} \]

Step 4: Simplify the Expression

Simplify the expression to find the slope: \[ m = \frac{0}{7} = 0 \]

Step 5: Interpretation of the Result

A slope of 0 indicates that the line is horizontal, meaning there is no rise over the run between the two points.

Key Concepts

Linear Equations

Linear equations are fundamental in mathematics and form the basis of many real-world applications. They describe straight lines on a graph and can be used to model various situations involving relationships between variables. Each linear equation can be written in the general form:

• Standard Form: \( ax + by = c \)
• Slope-Intercept Form: \( y = mx + b \)

Here, \( a \), \( b \), and \( c \) are constants, \( x \) and \( y \) are variables, and \( m \) represents the slope of the line, while \( b \) stands for the y-intercept. The slope \( m \) indicates how steep the line is and the direction in which it rises or falls as it moves from left to right. A horizontal line, where the slope is zero, signifies a linear equation like \( y = b \), which means the value of \( y \) is constant, as was seen in the exercise example.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to use algebraic equations to describe geometric figures and their properties. By defining points with coordinates (like \((x_1, y_1)\) and \((x_2, y_2)\)), it becomes possible to perform mathematical calculations directly related to geometry.

• Each point on a plane is determined by an ordered pair of numbers called coordinates.
• The first number indicates the point's horizontal position (x-coordinate), while the second number indicates the vertical position (y-coordinate).

Coordinate geometry is especially useful for determining the distance between points, the midpoint of a segment, and the slope of a line connecting two points. As in our exercise example, we used these principles to find the slope of a line by analyzing the coordinates provided. Understanding this relationship is crucial in linking algebraic equations with geometric representation.

Graphing Lines

Graphing lines is a visual way to understand equations and the relationships they represent. This process involves plotting points on a graph and connecting them to see how they form a line. Each line graph can provide insights into data trends or mathematical functions. Start by plotting individual points calculated from a linear equation. Once you have at least two points (as in our given exercise), connect them with a straight line to graph the equation. Here are some key tips:

• The slope of the line informs us how steep the line will be. A positive slope rises upward, whereas a negative slope falls downward.
• A horizontal line indicates a slope of zero, which means the line doesn't rise or fall as it moves horizontally, demonstrating constant y-values.
• The y-intercept is where the line crosses the y-axis, found at \(b\) in \(y = mx + b\).

Graphing lines can simplify complex equations, allowing us to interpret and predict behaviors visually.