91Ó°ÊÓ

The slope-intercept form of a linear equation is represented as \( y = mx + b \). This form is particularly useful because it gives us immediate insights into two key components of a linear equation: the slope and the y-intercept.

The letter \( m \) indicates the slope of the line. The slope measures the "steepness" of the line, showing how much \( y \) changes for a given change in \( x \). If the slope is positive, the line inclines upwards; if negative, it declines downwards. For example, in the equation we worked with, \( y = -\frac{1}{2}x + 3 \), the slope is \(-\frac{1}{2}\). This means for every 2 units the line runs horizontally to the right, it falls 1 unit.

• This equation was rearranged from its original form, \( \frac{1}{2}x + y = 3 \), to fit the slope-intercept form by isolating \( y \).
• The constant \( b \) in the equation (\( b = 3 \)) indicates where the line crosses the y-axis.

By using this form, you can quickly draw a graph without needing to calculate many points, simplifying the graphing process dramatically.

The y-intercept of a line is the point where the line crosses the y-axis. This happens when the value of \( x \) is equal to zero.

To find the y-intercept, you simply look at the constant term \( b \) in the slope-intercept form \( y = mx + b \). In our example, the equation \( y = -\frac{1}{2}x + 3 \) tells us directly that the y-intercept is 3.
This means that when \( x = 0 \), \( y \) equals 3. Therefore, the coordinate is \( (0, 3) \).

• The beauty of the y-intercept is that it's easily visible from the equation when in slope-intercept form.
• It's also crucial for rapidly graphing a line, as it provides one of the two points needed to draw a straight line on a graph.

The x-intercept is the point where the line crosses the x-axis, which occurs when \( y = 0 \).

To find the x-intercept from the equation \( y = -\frac{1}{2}x + 3 \), we set \( y \) to 0 and solve for \( x \):

• Start by reorganizing the equation: \( 0 = -\frac{1}{2}x + 3 \).
• Solving gives you \( \frac{1}{2}x = 3 \) by adding \( \frac{1}{2}x \) to both sides.
• Finally, multiply both sides by 2 to find \( x = 6 \).

This tells us that when the line hits the x-axis, it does so at the point \( (6, 0) \). Finding the x-intercept is crucial for completing the graph.
Using both x- and y-intercepts, you have two points which are sufficient to draw the linear graph accurately.