Problem 18 - Algebra A Combined Function

Question

Write an equation of the line passing through the given points. Write the equation in standard form \(A x+B y=C\). See Example 2 . \(\left(\frac{1}{2},-\frac{1}{4}\right)\) and \(\left(\frac{3}{2}, \frac{3}{4}\right)\)

Step-by-Step Solution

Answer

The equation of the line is \(x - 2y = 1\).

Step 1: Find the Slope

To find the slope of the line passing through two points, the formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the given points: \(x_1 = \frac{1}{2}, y_1 = -\frac{1}{4}\) and \(x_2 = \frac{3}{2}, y_2 = \frac{3}{4}\). \[m = \frac{\frac{3}{4} - \left(-\frac{1}{4}\right)}{\frac{3}{2} - \frac{1}{2}} = \frac{\frac{3}{4} + \frac{1}{4}}{1} = \frac{1}{2}\]. Therefore, the slope is \(\frac{1}{2}\).

Step 2: Find the y-intercept (b)

Using the slope-intercept form \(y = mx + b\), substitute one of the points into the equation to find \(b\). Using point \(\left(\frac{1}{2}, -\frac{1}{4}\right)\): \(-\frac{1}{4} = \frac{1}{2} \cdot \frac{1}{2} + b\). Simplifying gives \(-\frac{1}{4} = \frac{1}{4} + b\). Thus, \(b = -\frac{1}{4} - \frac{1}{4} = -\frac{1}{2}\).

Step 3: Write the Equation in Slope-Intercept Form

With the slope \(m = \frac{1}{2}\) and y-intercept \(b = -\frac{1}{2}\), the equation in slope-intercept form is \(y = \frac{1}{2}x - \frac{1}{2}\).

Step 4: Convert to Standard Form

To convert \(y = \frac{1}{2}x - \frac{1}{2}\) into standard form \(Ax + By = C\), first eliminate the fractions by multiplying the entire equation by 2: \(2y = x - 1\). Rearrange to get \(x - 2y = 1\). This is the equation in standard form.

Key Concepts

Slope of a Line

The slope of a line is a measure of how steep the line is. It tells us how much the line rises or falls as it moves from left to right. The formula for finding the slope, often represented by the letter \(m\), when given two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line, is:
\[m = \frac{y_2 - y_1}{x_2 - x_1}.\]This formula calculates the vertical change ("rise") divided by the horizontal change ("run").
For example, using the points \(\left(\frac{1}{2}, -\frac{1}{4}\right)\) and \(\left(\frac{3}{2}, \frac{3}{4}\right)\)from the exercise, we compute:

• Vertical change: \(\frac{3}{4} - (-\frac{1}{4}) = \frac{3}{4} + \frac{1}{4} = 1\)
• Horizontal change: \(\frac{3}{2} - \frac{1}{2} = 1\)

The slope \(m\) is then \(\frac{1}{1} = \frac{1}{2}\), meaning the line rises \(\frac{1}{2}\) unit for every 1 unit it moves to the right.

Slope-Intercept Form

The slope-intercept form of a line's equation shows both the slope and the y-intercept clearly, which is incredibly useful for quickly graphing or understanding the line's behavior. It is written as:
\[y = mx + b\]Where:

• \(m\) is the slope of the line.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

From our exercise, we calculated the slope \(m = \frac{1}{2}\), and by using one of the points, we found the y-intercept \(b = -\frac{1}{2}\).
Substituting these values into the slope-intercept form, we get:
\[y = \frac{1}{2}x - \frac{1}{2}.\]This form makes it easy to see the line rises by \(\frac{1}{2}\) for every step to the right, and it crosses the y-axis at \(-\frac{1}{2}\).

Standard Form of a Line

The standard form of a line's equation presents the line in a straightforward, integer-based manner. It is written as:
\[Ax + By = C\]Where \(A\), \(B\), and \(C\) are integers, and \(A\) should ideally be a positive integer. To convert from slope-intercept form to standard form, sometimes it is necessary to eliminate fractions and rearrange terms.
Starting with the slope-intercept form from our example, \(y = \frac{1}{2}x - \frac{1}{2}\), we:

• Eliminate fractions by multiplying the whole equation by 2: \(2y = x - 1\)
• Rearrange to present in standard form: \(x - 2y = 1\)

The final equation, \(x - 2y = 1\), now clearly shows the relationship between \(x\) and \(y\) with integer coefficients, useful for many mathematical and real-world applications.