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Slopes are an essential concept when analyzing linear equations. The slope of a line is a measure of its steepness and is usually denoted by the letter 'm'.
In the slope-intercept form of a linear equation, which is given by \( y = mx + b \), 'm' represents the slope. The slope can also be understood as the ratio of the rise (change in y) over the run (change in x) between two points on a line.
A positive slope means the line goes upward as we move from left to right. Conversely, a negative slope means the line goes downward as we move from left to right.

For example, in the equation \( y = \frac{3}{2}x - 4 \), the slope is \( \frac{3}{2} \). This positive slope means that for each unit increase in x, y will increase by \( \frac{3}{2} \). In the equation \( y = -\frac{2}{3}x + 2 \), the slope is \( -\frac{2}{3} \). This negative slope means that for each unit increase in x, y will decrease by \( \frac{2}{3} \).

Understanding slopes is important when determining the relationships between different lines, which brings us to the concepts of parallel and perpendicular lines.

The y-intercept is another core concept in understanding linear equations. It indicates the point where a line crosses the y-axis.
In the slope-intercept form \( y = mx + b \), 'b' represents the y-intercept. This is the value of y when x is zero.

For example, in the equation \( y = \frac{3}{2}x - 4 \), the y-intercept 'b' is -4. This means the line crosses the y-axis at the point (0, -4). Similarly, in the equation \( y = -\frac{2}{3}x + 2 \), the y-intercept is 2. This means the line crosses the y-axis at the point (0, 2).

Knowing the y-intercept helps you graph the line quickly and understand its position relative to the origin. Combining the knowledge of slopes and y-intercepts aids in fully grasping the orientation and position of a line in a coordinate plane.

Lines can have special relationships based on their slopes.
* **Parallel Lines:** These lines have the same slope and will never intersect. For example, if two lines have slopes \( m_1 \) and \( m_2 \) and \( m_1 = m_2 \), then the lines are parallel.

* **Perpendicular Lines:** These lines intersect at a right angle (90 degrees). Their slopes are negative reciprocals of each other. For instance, if the slope of one line is \( m_1 \) and the slope of the other line is \( m_2 \), then the lines are perpendicular if \( m_1 = -\frac{1}{m_2} \).

In our exercise, we found the slopes of the two lines to be \( \frac{3}{2} \) and \( -\frac{2}{3} \). Since \( \frac{3}{2} = -\frac{1}{-\frac{2}{3}} \), these lines are perpendicular. Recognizing these relationships helps in understanding the geometry of lines on a coordinate plane.