Problem 9 - A Graphical Approach to College Algebra

Question

Write the slope-intercept form of the line that passes through the given point with slope \(m .\) Do not use a calculator. Through \(\left(\frac{1}{4}, \frac{2}{3}\right), m=\frac{1}{2}\)

Step-by-Step Solution

Answer

The line's equation is \( y = \frac{1}{2}x + \frac{13}{24} \).

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept of the line.

Step 2: Plug in the Slope

Since the slope \( m \) is given as \( \frac{1}{2} \), substitute \( m = \frac{1}{2} \) into the equation. This gives us \( y = \frac{1}{2}x + b \).

Step 3: Substitute the Given Point into the Equation

Use the point \( \left(\frac{1}{4}, \frac{2}{3}\right) \) to substitute \( x = \frac{1}{4} \) and \( y = \frac{2}{3} \) into the equation \( y = \frac{1}{2}x + b \). This results in the equation \( \frac{2}{3} = \frac{1}{2}\left(\frac{1}{4}\right) + b \).

Step 4: Simplify the Equation

Calculate \( \frac{1}{2} \times \frac{1}{4} \) which equals \( \frac{1}{8} \). Then substitute back into the equation to get \( \frac{2}{3} = \frac{1}{8} + b \).

Step 5: Solve for b

Subtract \( \frac{1}{8} \) from \( \frac{2}{3} \) to isolate \( b \). Find a common denominator for the fractions, which is 24. Thus, \( \frac{2}{3} = \frac{16}{24} \) and \( \frac{1}{8} = \frac{3}{24} \). Therefore, \( b = \frac{16}{24} - \frac{3}{24} = \frac{13}{24} \).

Step 6: Write the Final Equation

Insert the calculated value of \( b = \frac{13}{24} \) back into the slope-intercept form. The equation of the line is \( y = \frac{1}{2}x + \frac{13}{24} \).

Key Concepts

Equation of a Line

When we refer to the "equation of a line," we're often talking about a way to mathematically describe a straight line on a graph. The most popular form used is the slope-intercept form. This form is particularly useful because it directly shows the slope and the y-intercept, which are key characteristics needed to graph a line.

The general formula for the slope-intercept form is:

• \( y = mx + b \)

Here, \( m \) stands for the slope of the line, and \( b \) represents the y-intercept, where the line crosses the y-axis. Once you know these values, you can easily draw the line or understand how it behaves in a graph.

When given a point and a slope, you can plug these values into the formula to find the equation of the line, as described in the original exercise. This allows you to predict other points on the line or graph it accurately.

Slope

The slope of a line is a measure that tells us how steep the line is. Mathematically, it's defined as the "rise over run"—how much the line goes up or down for each unit of horizontal movement to the right.

To calculate the slope, you can use the formula:

• \( m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1} \)

This can come in handy if you're given two points on a line, but when you're provided a slope directly (like \( m = \frac{1}{2} \) in the original exercise), you can substitute it directly into the slope-intercept equation.

A positive slope means the line goes upwards from left to right. A negative slope indicates it goes downwards. In practical terms, the given slope in the exercise illustrates how the line rises by 1 unit for every 2 units it moves to the right.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. This is an important feature because it provides a starting point for drawing the line on a graph.

In the slope-intercept form, \( b \) is the y-intercept:

• \( b \) represents the y-coordinate when the x-coordinate is zero (\( x = 0 \))

From our original exercise, after finding the slope \( m \) and plugging the point into the equation, we solve for \( b \) to find the y-intercept.

Understanding the y-intercept helps you graph the equation or verify points that lie on the line. If you know the y-coordinate where your line hits the y-axis, you can draw the line more accurately and understand its position in the graph.