Problem 9 - College Algebra with Modeling and Visualization

Question

If possible, find the slope of the line passing through each pair of points. $$ (-1,4),(5,-2) $$

Step-by-Step Solution

Answer

The slope of the line is \(-1\).

Step 1: Identify the given points

We are given two points: \((-1, 4)\) and \((5, -2)\).

Step 2: Recall the formula for the slope

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

Step 3: Substitute the points into the formula

For our points \((-1, 4)\) and \((5, -2)\), we substitute: \(x_1 = -1\), \(y_1 = 4\), \(x_2 = 5\), and \(y_2 = -2\) into the slope formula. So, \[ m = \frac{-2 - 4}{5 - (-1)} \]

Step 4: Calculate the differences

Calculate \(y_2 - y_1\) and \(x_2 - x_1\). We have \(-2 - 4 = -6\) and \(5 - (-1) = 5 + 1 = 6\).

Step 5: Simplify the slope fraction

Now substitute the differences into the slope formula: \[ m = \frac{-6}{6} \] Simplify the fraction to get \(m = -1\).

Key Concepts

Coordinate Geometry

Coordinate Geometry is a branch of geometry where the position of points on a plane is described using an ordered pair of numbers. These numbers are often called coordinates. Every point on a coordinate plane is represented as

• Coordinates are written as two numbers in a pair, usually

\((x, y)\), where \(x\) is the position along the horizontal axis, and \(y\) is the position along the vertical axis.
The two points provided in the exercise are

• The beauty of using coordinates is that they help us
• i.e.,

1. describe the locations of points clearly and precisely.
   This is extremely useful for solving problems in geometry and algebra, such as finding
2. \((-1, 4)\) and \((5, -2)\).

the slope of a line. Coordinate geometry forms the foundational understanding necessary to

• Coordinate systems like this
• engineers

solve more complex geometric problems and bridge the gap to algebra.support teachers, scientists, and coordinate geometry work with the Cartesian plane, which is fundamental for grid-based navigation and mapping.

Slope Formula

The slope of a line is a measure of how steep the line is. It’s calculated as a ratio that describes the vertical change between two points (often referred to as the 'rise') over the horizontal change (the 'run').

• This is an essential concept in mathematics, especially in Coordinate Geometry, that helps us understand and describe the inclination of lines on a graph.

By using the slope formula, \( m = \frac{y_2 - y_1}{x_2 - x_1} \), we can determine this characteristic of a line passing through two given points, \((x_1, y_1)\) and \((x_2, y_2)\). This formula is handy in both theoretical math problems and real-life applications such as

• understanding how fast something is increasing or decreasing.
• In our exercise,
  Substitute the values

\((5, -2)\) to find the slope.
we used the points \((-1, 4)\) and Then, we calculated:

• the slope as \(m = -1\).
• The value of
• negative slopes indicate a downward trend from left to right.
  

The slope is negative, which shows that the line decreases and indicates how steeply the decline occurs.

Linear Equations

Linear equations are equations that make a straight line when graphed on a coordinate plane. A common form of a linear equation is the slope-intercept form, represented mathematically as \(y = mx + b\), where:

• \(m\) represents the slope of the line.
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

Knowing the slope of a line helps us create its equation. For instance,

• this allows us to easily graph it or understand its behavior within larger mathematical models.
• Through substituting the slope value back into such equations,
  using the slope we calculated, \(m = -1\), we can construct the linear equation that describes the line passing through any specific pair of points.

For our example:

• considering a point on the line, we can find the equation by substituting the slope and coordinates of one point into the slope-intercept form.
  Linear equations are not only fundamental in academia but also in everyday scenarios
• which include planning budgets, designing architecture, and understanding trends and relationships within data.