Problem 3 - Algebra A Combined Function

Question

Find the slope of the line that passes through the given points. See Examples 1 and 2. $$ (1,4) \text { and }(5,3) $$

Step-by-Step Solution

Answer

The slope of the line is \(-\frac{1}{4}\).

Step 1: Identify the formula for finding slope

To find the slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]where \(m\) is the slope of the line.

Step 2: Substitute the coordinates into the formula

Plug the given coordinates \((x_1, y_1) = (1, 4)\) and \((x_2, y_2) = (5, 3)\) into the slope formula:\[m = \frac{3 - 4}{5 - 1}\]

Step 3: Calculate the difference in coordinates

Calculate the numerator \(3 - 4 = -1\) and the denominator \(5 - 1 = 4\).

Step 4: Compute the slope

Using the differences calculated, find the slope by dividing:\[m = \frac{-1}{4}\]The slope of the line is \(-\frac{1}{4}\).

Key Concepts

Linear Equations

Linear equations form the backbone of algebra and are essential for understanding many mathematical concepts. A linear equation represents a straight line when plotted on a coordinate graph. It typically takes the form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. This equation is called "linear" because its graph is a straight line.

Understanding the components:

• The slope (\(m\)) indicates how steep the line is and in which direction it points. A positive slope means the line rises as you move from left to right, while a negative slope indicates it falls.
• The y-intercept (\(b\)) is where the line crosses the y-axis. This is the value of \(y\) when \(x = 0\).

Linear equations are used extensively in various fields like physics, economics, and engineering to model relationships between variables. They are easy to solve, predict outcomes, and graph, making them a versatile tool in both academic and practical scenarios.

Coordinate Geometry

Coordinate geometry, also known as analytical geometry, is a branch of mathematics that uses algebraic formulas to describe geometric figures and their properties. It allows us to find the slope of a line, which is a measure of how much the line inclines from the horizontal.

Geometry in a coordinate plane:

• Points are defined by pairs of numbers in an ordered fashion, such as \((x, y)\), which represent their horizontal and vertical positions.
• The distance and direction between points can be calculated, determining lines, segments, and angles.
• By using this system, you can solve geometric problems using algebraic methods, which creates a powerful linkage between algebra and geometry.

Coordinate geometry allows us to visualize algebraic equations and offers a visual representation of relationships, providing a clearer understanding of concepts.

Algebraic Formulae

Algebraic formulae are tools that simplify the process of solving equations and problems in mathematics. They provide a structured way to carry out calculations with specific steps and rules. In finding the slope, the formula \[m = \frac{y_2 - y_1}{x_2 - x_1}\] is an algebraic method used to quantify the steepness of a line passing through two points.

Breaking down the formula:

• \(y_2 - y_1\) represents the change in the vertical direction between two points – known as the "rise."
• \(x_2 - x_1\) indicates the change in the horizontal direction – called the "run."
• Dividing the "rise" by the "run" (\(\frac{\text{rise}}{\text{run}}\)) gives the slope, expressing how much \(y\) changes for a unit change in \(x\).

Algebraic formulae like the slope formula allow us to make logical progressions and predictions in both theoretical and real-world contexts. They condense complex relationships into manageable calculations, simplifying understanding and application.