91Ó°ÊÓ

Linear equations are fundamental elements in algebra. They are equations that make a straight line when graphed on a coordinate plane. The basic form of a linear equation is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.
Understanding linear equations is crucial because they describe relationships with constant rates of change. These equations inherently have only variables raised to the first power and no terms multiplied or divided by each other.
In the given exercise, the linear equation is \(8x - 2y = 1\). To work comfortably with equations like this, we often rearrange them into a more useful form known as the slope-intercept form by solving for \(y\). This leads to simpler manipulation and graphing.

The slope-intercept form is a specific format of writing linear equations. It is represented as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) denotes the y-intercept, where the line crosses the y-axis.
One of the big advantages of using the slope-intercept form is its straightforwardness in identifying the line's slope. It tells us how steep the line is and in what direction it runs. A positive slope means the line ascends from left to right, while a negative slope means it descends.
In the given equation, after simplifying and solving for \(y\), we converted \(8x - 2y = 1\) into slope-intercept form, resulting in \(y = 4x - \frac{1}{2}\). This clearly shows us the slope \(m\) is 4, easily providing the rate of change.

Solving for \(y\) in a linear equation involves rearranging the equation to get \(y\) on one side. This method provides us with the slope-intercept form, allowing easy extraction of crucial line properties.
Let's break down the steps used in the exercise:

• Start with the original equation: \(8x - 2y = 1\).
• Simplify by dividing each term by 2, yielding: \(4x - y = \frac{1}{2}\).
• Isolate \(y\) by moving \(4x\) to the other side: \(y = 4x - \frac{1}{2}\).

These steps isolate \(y\) and provide an easy-to-use form of the equation. Solving for \(y\) is always a critical step in determining the slope and interpreting the linear relationship in a graphical context.