91Ó°ÊÓ

In linear equations, the slope-intercept form is a way to organize the equation in a manner that's easy to interpret and graph. This format is expressed as \( y = mx + b \). Here, \( m \) represents the slope of the line, which indicates its steepness, and \( b \) is the y-intercept, the point where the line crosses the y-axis.

To convert any linear equation to the slope-intercept form, you need to solve for \( y \). Taking our exercise, the equation is \( 6x - 5y - 20 = 0 \). By rearranging the terms, we first subtract \( 6x \) from both sides to get \( -5y = -6x + 20 \). This helps in isolating the \( y \)-term.

Then, you divide everything by \( -5 \) to isolate \( y \) fully, resulting in \( y = \frac{6}{5}x - 4 \). Now, the equation is in slope-intercept form, making it clear that \( m = \frac{6}{5} \) and \( b = -4 \). This transformation helps us easily identify the slope and y-intercept, crucial for graphing the linear equation.

Graphing linear functions is all about representing the equation visually on a coordinate plane. Once the equation \( y = \frac{6}{5}x - 4 \) is in slope-intercept form, plotting becomes straightforward. Here's how you can do it:

• Start by drawing a coordinate plane and mark your axes.
• Place your first point on the graph where the line will cross the y-axis, that is the y-intercept (0, -4ect).

A slope of \( \frac{6}{5} \) means for every 5 units moved along the x-axis, the line rises by 6 units. To plot this:

• From the point (0, -4), move 5 units to the right (positive direction on the x-axis).
• Then, move 6 units up (positive direction on the y-axis).

Once you have multiple points plotted using this method, draw a straight line through these points, extending it across the graph. This line represents the linear equation, confirming the relationship between the slope, y-intercept, and its graphical representation.

The slope and y-intercept are two fundamental components that describe a linear function. Understanding these helps you predict the behavior of the line and makes it easier to construct it visually.

**Slope (\( m \)):** This determines how steep or flat the line is and the direction it goes. A positive slope, like \( \frac{6}{5} \) from our equation, means the line rises as it moves from left to right. Conversely, a negative slope indicates a line falling.

**Y-intercept (\( b \)):** This represents where the line meets the y-axis. In \( y = \frac{6}{5}x - 4 \), the y-intercept is \(-4\). It tells us that irrespective of \( x \), the line will always cross the y-axis at this point.

Together, the slope tells us how to adjust our position on the graph starting from the y-intercept. Understanding these two elements is key to mastering linear equations and graphing efficiently.