91Ó°ÊÓ

The slope-intercept form of a linear equation is one of the simplest and most frequently used forms. It is written as:
\( y = mx + b \)
Where:

• \( y \) represents the value of the dependent variable (often on the vertical axis).
• \( x \) is the independent variable (usually on the horizontal axis).
• \( m \) denotes the slope of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

To convert a given equation into slope-intercept form, you need to isolate \( y \) on one side of the equation. For example, the equation \( 5x - 2y = 11 \) can be rearranged by solving for \( y \). First, subtract \( 5x \) from both sides:
\[ -2y = -5x + 11 \]
Then, divide every term by \( -2 \):
\[ y = \frac{5}{2} x - \frac{11}{2} \].
Now it's in slope-intercept form with slope \( \frac{5}{2} \) and y-intercept \( -\frac{11}{2} \).

The slope of a line measures its steepness and direction. Calculated as the ratio of the vertical change to the horizontal change between two points on the line, the slope \( m \) is given by:
\[ m = \frac{ \text{rise} }{ \text{run} } \]
To find the slope of a line from its equation in slope-intercept form, simply identify the coefficient of the \( x \) term. For \( y = 5x - 7 \), the slope \( m \) is \( 5 \).

When comparing the slopes of two lines:

• If their slopes are equal (\fastmath{//} \( m1 = m2 \)), the lines are parallel.
• If the product of their slopes is \( -1 \) (\fastmath{//} \( m1 \times m2 = -1 \)), the lines are perpendicular.
• If the slopes are different and the product is not \( -1 \), the lines intersect but are neither parallel nor perpendicular.

For example, consider the slopes from the provided equations:
\[ m_1 = \frac{5}{2} \]
and
\[ m_2 = 5 \].
Since \( \frac{5}{2} eq 5 \), these lines are not parallel.

Equations of lines can be represented in various forms, with the slope-intercept form being one of the most common. Another frequently used form is the point-slope form:
\[ y - y_1 = m(x - x_1) \]
Here:

• \( (x_1, y_1) \) is a known point on the line.
• \( m \) is the slope of the line.

For problems involving determining if lines are parallel, converting equations to slope-intercept form is essential as it directly reveals the slope.
For instance, given the equations
\[ 5x - 2y = 11 \] and
\[ 5x - y = 7 \],
converting these to slope-intercept form provides:

For the first equation: \[ y = \frac{5}{2} x - \frac{11}{2} \]
And for the second equation: \[ y = 5x - 7 \]
By comparing their slopes \( \frac{5}{2} \) and \( 5 \), you can determine they aren't parallel.
So when dealing with parallelism, focus on the slopes and ensure both lines are in a comparable form, typically the slope-intercept form.