Problem 47 - A Graphical Approach to Precalculus with Limits

Question

Find the slope (if defined) of the line that passes through the given points. $$\left(\frac{1}{2},-\frac{2}{3}\right) \text { and }\left(-\frac{3}{4}, \frac{1}{6}\right)$$

Step-by-Step Solution

Answer

The slope of the line is \(-\frac{2}{3}\).

Step 1: Identify the Formula for Slope

The formula to find the slope \(m\) of a 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 Values to Variables

Identify and assign the coordinates from the points to the variables: \(x_1 = \frac{1}{2}, y_1 = -\frac{2}{3}, x_2 = -\frac{3}{4}, y_2 = \frac{1}{6}\).

Step 3: Subtract the Y-values

Calculate the difference in the \(y\)-coordinates: \[y_2 - y_1 = \frac{1}{6} - \left(-\frac{2}{3}\right)\] Convert \( \frac{1}{6} \) and \( \frac{2}{3} \) to have a common denominator: \( \frac{1}{6} = \frac{1}{6} \) and \( -\frac{2}{3} = -\frac{4}{6} \). So, \[y_2 - y_1 = \frac{1}{6} + \frac{4}{6} = \frac{5}{6}\].

Step 4: Subtract the X-values

Calculate the difference in the \(x\)-coordinates: \[x_2 - x_1 = -\frac{3}{4} - \frac{1}{2}\]Convert \(-\frac{3}{4}\) and \(\frac{1}{2}\) to a common denominator:\(-\frac{3}{4} = -\frac{3}{4}\) and \(\frac{1}{2} = \frac{2}{4}\).So, \[ x_2 - x_1 = -\frac{3}{4} - \frac{2}{4} = -\frac{5}{4}.\]

Step 5: Calculate the Slope

Use these differences to find the slope: \[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\frac{5}{6}}{-\frac{5}{4}} = -\frac{5}{6} \times \frac{4}{5} = -\frac{20}{30} = -\frac{2}{3}.\] Thus, the slope of the line is \(-\frac{2}{3}\).

Key Concepts

Understanding Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a mathematical system that uses coordinates to uniquely identify points on a plane. This form of geometry simplifies the process of understanding geometric shapes and their properties by using algebraic expressions. The plane consists of two axes, usually labeled as the x-axis (horizontal) and the y-axis (vertical). From the origin, where these axes intersect, every point on the plane is defined by an ordered pair of numbers

• First number (x-coordinate): This number tells you how far a point is horizontally from the origin. A positive number means it's to the right, and a negative number means it's to the left.
• Second number (y-coordinate): This number indicates how far the point is vertically from the origin. Positive means up, while negative means down.

In the context of lines, coordinate geometry allows us to calculate key metrics, such as the slope. Finding the slope is crucial for understanding the direction and steepness of the line. A positive slope means the line rises as it moves from left to right, whereas a negative slope means it falls.

Mastering Fraction Operations

When dealing with coordinate geometry, fraction operations often come into play, especially when the coordinates involve fractions. Understanding these operations is crucial to solve geometry problems accurately. Here are some essential fraction operations you'll frequently encounter:

• Adding/Subtracting Fractions: To add or subtract fractions, they must first have a common denominator. This means both fractions should have the same bottom number, representing equal parts of the same whole. You adjust fractions to common denominators by finding the least common multiple of both denominators.
• Multiplying Fractions: Fortunately, multiplying fractions is simpler. You just multiply the numerators together and the denominators together. Then simplify the resulting fraction if possible.
• Dividing Fractions: Division is done by multiplying by the reciprocal of the divisor. Essentially, "flip" the second fraction and multiply.

Recognizing these operations enables you to handle problems involving lines and slopes effectively, such as computing changes in coordinates or simplifying equations.

Exploring Linear Equations

Linear equations stand at the core of both algebra and coordinate geometry. These equations describe lines and can be written in several forms, the most common being the slope-intercept form, given by:\[ y = mx + c \]Where:

• \( m \) is the slope or gradient of the line. It tells you how steep the line is.
• \( c \) is the y-intercept, which is the value where the line crosses the y-axis.

Linear equations are a powerful tool because they can describe all lines on a coordinate plane. By using a linear equation, you can predict the value of one variable if the other is known. Understanding how to manipulate linear equations, for example by altering the slope or intercept, enables you to model real-world relationships and solve various problems in mathematics.