Problem 24 - College Algebra with Corequisite Support

Question

For the following exercises, find the slope of the line that passes through the two given points. (2,4) and (4,10)

Step-by-Step Solution

Answer

The slope of the line is 3.

Step 1: Identify the Formula for Slope

The formula to find the slope \( m \) of the 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: Assign Coordinates to Variables

Assign the coordinates of the given points to the formula variables:\((x_1, y_1) = (2, 4)\) and \((x_2, y_2) = (4, 10)\).

Step 3: Substitute and Calculate Differences

Substitute the points into the slope formula:\[ m = \frac{10 - 4}{4 - 2} \] Calculate the differences: \[ y_2 - y_1 = 10 - 4 = 6 \] \[ x_2 - x_1 = 4 - 2 = 2 \]

Step 4: Compute the Slope

Now, use the differences to find the slope: \[ m = \frac{6}{2} = 3 \] Thus, the slope \( m \) of the line is 3.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses coordinates to describe geometric figures and properties. It allows us to transform geometrical shapes and lines into algebraic expressions, which can be solved using mathematical equations. By using coordinates, we can easily find and describe properties of geometric shapes such as lines, circles, and different types of curves.

In coordinate geometry, points are typically described using two numbers, called coordinates, often represented as \( (x, y) \). These numbers represent a location on a two-dimensional plane.

• The first number, \( x \), is called the x-coordinate or abscissa. It indicates how far a point is along the horizontal axis.
• The second number, \ (y) \, is known as the y-coordinate or ordinate. It shows how far the point is along the vertical axis.

Coordinate geometry is highly useful for solving problems involving distances, midpoints, and especially the slope of a line, as it involves the use of the coordinates of points to perform calculations.

Linear Equations

Linear equations are mathematical expressions that represent straight lines when plotted on a coordinate plane. These equations are usually written in the form \( y = mx + c \), where \( m \) represents the slope and \( c \) is the y-intercept, which is the point where the line crosses the y-axis.

The slope \( m \) indicates the steepness and direction of a line. A positive slope means the line is ascending as it moves from left to right, while a negative slope indicates a descending line. If the slope is zero, the line is horizontal and, if undefined, it is a vertical line.

• Linear equations can help predict outcomes or understand relationships between variables.
• To formulate a linear equation, you sometimes need two key pieces of information: the slope and any specific point on the line.

Understanding linear equations is crucial in many fields such as physics, economics, and engineering, as they model real-world problems effectively and can be solved using various algebraic methods.

Mathematical Formula

A mathematical formula is a rule or relationship expressed through symbols, numbers, and operators. In essence, a formula represents a concise way to express information. It allows consistent calculation by providing a pathway to find a solution or to describe how things relate.

In this context, we're interested in the formula for finding the slope of a line. The slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) is pivotal to coordinate geometry and linear equations. This formula provides a method to determine how steep a line is between two points. By calculating the vertical change \( (y_2 - y_1) \) over the horizontal change \( (x_2 - x_1) \), we derive the line's slope.

• Formulas like these are essential in mathematics because they simplify complex ideas into manageable calculations.
• They also ensure accuracy and consistency across various computations.

Grasping the use of mathematical formulas such as the slope formula aids in the analysis of linear relationships and supports problem-solving across numerous mathematical problems.