Problem 58 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write an equation of the line that passes through the points. Use the slope- intercept form (if possible). If not possible, explain why and use the general form. Use a graphing utility to graph the line (if possible). $$\left(\frac{3}{4}, \frac{3}{2}\right),\left(-\frac{4}{3}, \frac{7}{4}\right)$$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form that passes through the points \(\left(\frac{3}{4}, \frac{3}{2}\right)\) and \(\left(-\frac{4}{3}, \frac{7}{4}\right)\) is \(y=\frac{1}{4}x + 1\).

Step 1: Calculate the Slope of the Line

The formula for calculating the slope, m, of a line when given two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). By substitution, we get \(m = \frac{\frac{7}{4} - \frac{3}{2}}{-\frac{4}{3} - \frac{3}{4}} = \frac{1}{4}\).

Step 2: Insert the Slope and a Point Into the Slope-Intercept Form

The slope-intercept form of a linear equation is \(y = mx + b\), where m is the slope, and b is the y-intercept. Substituting the values of one of the points and the calculated slope into the equation will enable us to find the value of b. Let's use the point \(\left(\frac{3}{4}, \frac{3}{2}\right)\). This gives us \(\frac{3}{2}= \frac{1}{4}*\frac{3}{4} + b\). Solving for b gives \(b = 1\).

Step 3: Write the Equation of the Line

Now that we have found values for m and b, we simply insert these values back into the slope-intercept form equation. The final equation of the line is \(y = \frac{1}{4}x + 1\).

Step 4: Graphing the Line

This is mathematical. Therefore, it's difficult to demonstrate here. However, to graph this line, one would need to start at the point where y = 1 (this is the y-intercept) and go up 1 unit and across 4 units (this is the slope) to plot the next point. Continue this until you have plotted several points, then draw a straight line through these points with a ruler or a straight edge. This creates a visual representation of the line described by the equation.

Key Concepts

Linear Equations

A linear equation is a type of equation that represents a straight line when graphed on a coordinate plane. These equations are usually written in the form of \( y = mx + b \), known as the slope-intercept form. The variables \( x \) and \( y \) represent the coordinates of any point on the line, \( m \) stands for the slope of the line, and \( b \) is the y-intercept—the point where the line crosses the y-axis.
Linear equations are extremely important in mathematics as they are used to model a variety of real-world situations. For instance, calculating cost based on a fixed rate and an initial fee can often be illustrated with a linear equation.

• In slope-intercept form, \( m \) indicates how inclined the line is, and it expresses the rise over run, or the change in y-values over the change in x-values between two points.
• The y-intercept, \( b \), provides the starting value of \( y \) when \( x \) is zero.

Understanding linear equations forms a foundation for more advanced algebra and geometry topics.

Graphing Lines

Graphing lines is a fundamental skill in algebra that involves plotting a linear equation on a coordinate plane to visually represent the relationship between two variables. To graph a line in slope-intercept form \( y = mx + b \), follow these steps:

• Start by identifying the y-intercept \( b \). This is the point where the line will intersect the y-axis. Plot this point on your graph.
• Use the slope \( m \), which is a fraction (\( \frac{rise}{run} \)), to determine your next point. From the y-intercept, move vertically by the "rise" (the numerator) and horizontally by the "run" (the denominator) to plot the next point.
• Draw a straight line through these points, extending in both directions, to complete the graph.

Visualizing a line in this manner helps in understanding how changes in the slope or y-intercept affect the position and steepness of the line. It also aids in solving and verifying solutions to linear equations.

Slope Calculation

Calculating the slope of a line is crucial in understanding how one variable affects another in a linear relationship. The slope \( m \) is the "steepness" or "incline" of the line and is calculated by comparing the vertical change (\( \Delta y \)) to the horizontal change (\( \Delta x \)) between two points on a line.
The formula for finding the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

• This formula tells us how many units the line rises (or falls) for each unit it moves horizontally. A positive slope means the line ascends from left to right, while a negative slope indicates it descends.
• If a line has a slope of zero, it is perfectly horizontal, indicating no vertical change as \( x \) changes.
• An undefined slope occurs in vertical lines, where \( x_1 = x_2 \), resulting in division by zero.

Knowing how to calculate and interpret slope is essential for writing linear equations and understanding the direct relationship between variables.