Problem 34 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. $$ \text { Passing through }\left(-\frac{7}{3}, \frac{5}{3}\right) \text { and }\left(\frac{5}{6},-\frac{7}{6}\right) $$

Step-by-Step Solution

Answer

The equation is \( y = -\frac{17}{19}x - \frac{8}{19} \).

Step 1: Understanding the Problem

We need to find the equation of a line in slope-intercept form, which is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, passing through two given points: \((-\frac{7}{3}, \frac{5}{3})\) and \((\frac{5}{6}, -\frac{7}{6})\).

Step 2: Calculating Slope

The slope \( m \) of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the given points, \( m = \frac{-\frac{7}{6} - \frac{5}{3}}{\frac{5}{6} - (-\frac{7}{3})} \).

Step 3: Simplifying the Slope

First, simplify the numerator: \(-\frac{7}{6} - \frac{5}{3} = -\frac{7}{6} - \frac{10}{6} = -\frac{17}{6}\). Then simplify the denominator: \(\frac{5}{6} + \frac{7}{3} = \frac{5}{6} + \frac{14}{6} = \frac{19}{6}\). Thus, the slope \( m = \frac{-\frac{17}{6}}{\frac{19}{6}} = -\frac{17}{19}\).

Step 4: Finding the Y-intercept

Using the point-slope form of a line equation \( y - y_1 = m(x - x_1) \) and one of our points \((-\frac{7}{3}, \frac{5}{3})\), substitute \( m = -\frac{17}{19} \): \( y - \frac{5}{3} = -\frac{17}{19}(x + \frac{7}{3}) \). Expanding gives \( y = -\frac{17}{19}x - \frac{119}{57} + \frac{5}{3} \).

Step 5: Simplifying Y-intercept

Combine the constants to find \( b \), the y-intercept. Convert \( \frac{5}{3} \) to \( \frac{95}{57} \), then combine with \(-\frac{119}{57}\): \( b = -\frac{119}{57} + \frac{95}{57} = -\frac{24}{57} \) (simplify to \( -\frac{8}{19} \)).

Step 6: Final Equation

Substitute \( m = -\frac{17}{19} \) and \( b = -\frac{8}{19} \) into the slope-intercept form \( y = mx + b \): \( y = -\frac{17}{19}x - \frac{8}{19} \).

Key Concepts

Algebra

Algebra is like the language that helps us understand relationships between numbers and variables. When we talk about finding the slope-intercept form in algebra, we refer to the equation of a line, expressed in the form \( y = mx + b \). Here, \( m \) is the slope of the line, which shows how steep the line is, and \( b \) is the y-intercept, representing the point where the line crosses the y-axis.

To find this form, you need two key pieces of information: the slope and the y-intercept. In problems like the one given, you often start by calculating the slope using two points on the line. From there, algebraic manipulation helps you find the y-intercept. Once both values are known, they are plugged into the slope-intercept equation.

Algebra simplifies this process, breaking it down into manageable steps and making it easier to work with various types of equations in mathematics. Understanding these basic concepts in algebra is crucial for addressing more complicated mathematical problems.

Line Equations

Line equations describe the geometric properties and course of a line on a graph. They are important in coordinate geometry as they give explicit instructions on how to plot the line on a coordinate grid. The slope-intercept form is particularly useful because it directly shows the slope and y-intercept.

Slope in Line Equations

The slope in the line equation represents the rise over run, or how much the line goes up or down as you move along it horizontally. This measurement of steepness tells you how closely the line parallels the x-axis.

Y-intercept in Line Equations

The y-intercept is significant because it shows where the line meets the y-axis. This point is crucial for graphing because it's often an easy starting point for plotting your line. Together, the slope and y-intercept allow for a complete understanding of the line's direction and placement on a graph.

Coordinate Geometry

Coordinate geometry, sometimes called analytical geometry, merges algebra and geometry into one field of mathematical study. By using a coordinate plane, points (such as those given in the exercise: \((-\frac{7}{3}, \frac{5}{3})\) and \((\frac{5}{6}, -\frac{7}{6})\)) can be precisely located and manipulated mathematically.

Utility of Coordinate Geometry

Coordinate geometry is essential in connecting the abstract calculations of algebra with tangible geometric shapes. It allows us to express geometric concepts such as lines and angles through equations and formulas, offering a dual perspective.

This method can combine algebraic techniques, like solving equations, with geometric understanding to derive meaningful information about the object being studied. In the case of lines, coordinate geometry allows you to compute directly how one point relates to another through forms like the slope-intercept equation, illustrating both the path and direction of the line as it moves across the plane.