91Ó°ÊÓ

The slope-intercept form is a popular way to express the equation of a straight line. This form is written as \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept. The y-intercept is the point where the line crosses the y-axis. This form is particularly useful because it makes it easy to see the slope and intercept directly from the equation.
To convert an equation to the slope-intercept form, you need to solve for \( y \). For example, given the equation \( x + 5y = 9 \), we can rearrange it by isolating \( y \):

• Subtract \( x \) from both sides to get \( 5y = -x + 9 \).
• Then, divide each term by 5: \( y = -\frac{1}{5}x + \frac{9}{5} \).

This transformed equation now clearly shows the slope \( m = -\frac{1}{5} \) and the y-intercept \( b = \frac{9}{5} \). This makes it easier to graph the line or compare it with other lines.

Another useful way to express the equation of a line is through the point-slope form. This form is especially handy when you know a point on the line and the slope. It is written as \( y - y_1 = m(x - x_1) \), where \( m \) is the slope, and \( (x_1, y_1) \) are the coordinates of the given point on the line.
In our exercise, we were given a point \((1,3)\) and knew the slope was \(-\frac{1}{5}\). Using point-slope form, we substitute these values in:

• Start with \( y - 3 = -\frac{1}{5}(x - 1) \).
• To simplify, distribute the slope: \( y - 3 = -\frac{1}{5}x + \frac{1}{5} \).
• Finally, solve for \( y \) by adding 3 to both sides: \( y = -\frac{1}{5}x + \frac{1}{5} + 3 \).
• Convert 3 to fraction: \( y = -\frac{1}{5}x + \frac{16}{5} \).

The point-slope form is flexible and easily converts to others, like the slope-intercept form.

Parallel lines offer an interesting relationship in geometry. Two lines are parallel if they have the same slope but different y-intercepts. This means parallel lines can go on infinitely without ever intersecting.
In the context of this exercise, we were dealing with a line parallel to \( x + 5y = 9 \). After converting the equation into slope-intercept form, we determined its slope as \(-\frac{1}{5}\).

• Since parallel lines share the same slope, any line parallel to the original has the slope \(-\frac{1}{5}\).
• The different y-intercept will depend on the specific point you use to derive this parallel line, as seen with our exercise.

Parallel lines are powerful in identifying lines that run alongside, which has practical implications, from railways to architecture. Understanding this concept helps us solve problems where direction, not position, is key.