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The first step in determining the slope of a line involves understanding its significance and identifying it in an equation. The slope is represented by the variable \( m \) in a linear equation of the form \( y = mx + b \). It measures the steepness or incline of the line on the graph. Essentially, it tells us how much \( y \) changes for a unit change in \( x \). A positive slope indicates the line rises as it moves from left to right, while a negative slope means the line falls.

For the equation given, \( 12x = 6y + 4 \), we need to rearrange it into the slope-intercept form to recognize \( m \). Once rearranged and simplified, the equation becomes \( y = 2x - \frac{2}{3} \). Here, the slope \( m \) is \( 2 \). This value indicates that for each unit increase in \( x \), \( y \) increases by \( 2 \).

To calculate the slope, you should:

• Rearrange the equation to isolate \( y \).
• Simplify the equation to the format \( y = mx + b \).
• Identify the coefficient of \( x \) as the slope \( m \).

The \( y \)-intercept is a key component in the slope-intercept form \( y = mx + b \) of a linear equation. It is represented by \( b \) and shows the point where the line crosses the \( y \)-axis. In simpler terms, it is the value of \( y \) when \( x \) is zero.

In the rearranged equation \( y = 2x - \frac{2}{3} \), the \( y \)-intercept \( b \) is \(-\frac{2}{3} \). This means that if you start at the point where the line crosses the \( y \)-axis, this value gives you the starting position of the line on the vertical axis. Understanding the \( y \)-intercept helps in graphing the line and quickly grasping where it lies on the graph.

To identify the \( y \)-intercept:

• Ensure the equation is in the \( y = mx + b \) form.
• Locate the constant term \( b \).
• Take note of its sign and magnitude as they affect the line's position on the graph.

Linear equations represent some of the most fundamental relationships between two variables, \( x \) and \( y \). These equations graph as straight lines in the coordinate plane and often take the form of \( y = mx + b \). This form is known as the slope-intercept form.

A linear equation expresses a constant rate of change. This constant rate is manifested in the slope \( m \), while the \( y \)-intercept \( b \) determines where the line intersects the \( y \)-axis.

Given the problem, rearranging the equation \( 12x = 6y + 4 \) into \( y = 2x - \frac{2}{3} \) allows us to determine both the slope and the \( y \)-intercept. Solving linear equations is a process of simplifying equations to identify these components, providing insight into how the line behaves on a graph.

When dealing with linear equations:

• Ensure you manage the terms carefully to rearrange them into the \( y = mx + b \) form.
• Use algebraic techniques such as addition, subtraction, multiplication, or division to isolate \( y \).
• Verify your final expression matches the standard form to accurately read values for \( m \) and \( b \).