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When dealing with linear equations, calculating the slope is an essential skill. The slope tells us how steep a line is and in which direction it goes. A positive slope, such as our example of \( \frac{2}{3} \), indicates that the line rises as you move to the right. Conversely, a negative slope would show a line that falls as you go from left to right.
To find the slope from an equation in the form \( y = mx + b \), look at the coefficient of the \( x \) term, which is \( m \). In our case, once we rearranged the equation to \( y = \frac{2}{3}x - 8 \), we can clearly see that the slope \( m \) is \( \frac{2}{3} \). This means for every three units you move horizontally, you move up two units vertically.
Remember that the slope is like a ratio or a fraction comparing the vertical change (rise) to the horizontal change (run).
Keep in mind:

• A slope of \( 0 \) means the line is perfectly horizontal.
• Vertical lines have an undefined slope because you can't divide by zero (infinite rise with zero run).

Identifying the y-intercept is another key part of understanding linear equations. The y-intercept is where the line crosses the y-axis and is denoted by the \( b \) in the equation form \( y = mx + b \).
In the line equation \( y = \frac{2}{3}x - 8 \), the y-intercept is \( -8 \). This tells us that when \( x \) is \( 0 \), \( y \) is \( -8 \). It means the point where the line hits the y-axis is at (0, -8).
Understanding the y-intercept:

• It gives you the starting point of the line when graphing.
• If \( b \) is positive, the y-intercept is above the origin; if negative, it's below.

By knowing the y-intercept, you can more easily graph the line and understand its general position on the graph.

Rearranging an equation is crucial for revealing information like the slope and y-intercept. A common form used in these situations is the slope-intercept form, \( y = mx + b \). This helps us quickly read off information about the line.
The original equation \( 2x - 3y = 24 \) needs to be rearranged to focus on \( y \). Here are the steps in detail:

• Subtract \( 2x \) from both sides to isolate \( y \)-terms:
  \(-3y = -2x + 24\).
• Divide every term by \(-3\) to solve for \( y \):
  \( y = \frac{2}{3}x - 8 \).

This rearranged equation now clearly reveals the slope \( \frac{2}{3} \) and y-intercept \( -8 \). Knowing how to rearrange equations is vital for solving and understanding linear algebraic problems effectively.
Equation rearrangement allows us to:

• Simplify expressions.
• Make it easier to comprehend the relationships between variables.
• Visualize the equation as a line with specific characteristics directly.

Mastering this skill means you'll be more confident in tackling a variety of math challenges.