Problem 40 - A Graphical Approach to College Algebra

Question

Find the slope (if defined) of the line that passes through the given points. $$(-2,3)\( and \)(-1,2) \quad$$

Step-by-Step Solution

Answer

The slope of the line is -1.

Step 1: Identify the points

The given points are \((-2, 3)\) and \((-1, 2)\). To find the slope, we identify these coordinates as \((x_1, y_1) = (-2, 3)\) and \((x_2, y_2) = (-1, 2)\).

Step 2: Use the slope formula

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 values

Substitute the values from the points into the slope formula: \[ m = \frac{2 - 3}{-1 - (-2)} \].

Step 4: Simplify the expression

Calculate the difference for the numerator and the denominator separately: \(2 - 3 = -1\) and \(-1 - (-2) = -1 + 2 = 1\). Thus, \[ m = \frac{-1}{1}. \]

Step 5: Calculate the slope

Simplify the fraction to find the slope: \(-1 \div 1 = -1\). Thus, the slope of the line is \(-1\).

Key Concepts

Slope Formula

In the study of coordinate geometry, understanding the slope of a line is a key concept. The slope tells us how steep a line is, or in other words, how much the line rises or falls as we move along it. The slope formula is our friend here, defined as - Slope (\( m \)) = (change in \( y \)-coordinates) / (change in \( x \)-coordinates)It measures the "rise" over "run". When given two points, say \((x_1, y_1)\)and \((x_2, y_2)\)in the plane, you can compute the slope \( m \)using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1}. \]This is a simple subtraction of y-coordinates divided by the subtraction of x-coordinates. It's a straightforward way to find how much one variable changes relative to another. The slope can be interpreted as:

• Positive slope: The line rises as it moves from left to right.
• Negative slope: The line falls as it moves from left to right.
• Zero slope: The line is horizontal.
• Undefined slope: The line is vertical.

This formula becomes a fundamental tool when dealing with linear equations and can be applied to numerous mathematical problems.

Coordinate Geometry

Coordinate geometry, sometimes referred to as analytic geometry, serves as the bridge between algebra and geometry. It allows mathematicians and students to translate geometric shapes into numerical data using a coordinate plane. In this system, each point is represented by a pair of numbers, known as coordinates, often described as \((x, y)\).By plotting these points on a two-dimensional plane, you can visualize various geometric shapes and understand spatial relationships. The coordinate plane is divided into quadrants:

• Quadrant I: Positive values of \(x\)and \(y\)
• Quadrant II: Negative \(x\)and positive \(y\)
• Quadrant III: Negative values for both \(x\)and \(y\)
• Quadrant IV: Positive \(x\)and negative \(y\)

By using coordinate geometry principles, you can solve complex geometric problems using algebra. Consider a line passing through points \((-2, 3)\)and \((-1, 2)\).Here, the geometric interpretation involves plotting these points and understanding the linear path between them. Coordinate geometry allows you to find the slope, midpoint, and even the equation of the line connecting them.

Algebra

Algebra is a core branch of mathematics that can solve many problems by manipulating equations and expressions. It often involves finding unknown values through various techniques and formulas. In the context of finding the slope of a line, algebraic manipulation is essential for calculating values and simplifying expressions.In algebra, you often use:

• Variables: Symbols like \(x\)and \(y\)that stand for numbers.
• Constants: Fixed numbers such as 2, -1, or 3.
• Operations: Actions such as addition, subtraction, multiplication, and division.
• Equations: Mathematical statements that show the equality between two expressions.

To clarify this, consider the example where you substitute point values into the slope formula. The steps of substituting \( y_2 = 2, y_1 = 3, x_2 = -1,\)and\( x_1 = -2 \)into the slope formula involve basic algebra: \[ m = \frac{2 - 3}{-1 - (-2)} = \frac{-1}{1} = -1. \]This simplification process involves subtraction and division, classic algebraic operations, leading to the final slope value of the line. Understanding algebra enables you to seamlessly solve these expressions and apply them to similar problems.