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Understanding the point-slope form of a linear equation is essential in algebra. It's a way of expressing a linear equation when you know a single point on the line \( (x_1,y_1) \) and the slope of the line, \( m \). The general formula is \( y - y_1 = m(x - x_1) \). In the context of the exercise, the equation \( y-(-2)=\frac{3}{2}(x-1) \) already gives us the slope, \( m=\frac{3}{2} \), and a point \( (1,-2) \) that lies on the line.

When interpreting the point-slope equation, identify the values and their meaning. In our case, the coordinates \( (1, -2) \) tell us where the line passes through, and the slope \( \frac{3}{2} \) tells us the line's steepness and direction. By rearranging the equation, we can compare other forms and derive additional characteristics of the line, such as the y-intercept in the slope-intercept form.

The slope-intercept form, \( y = mx + b \), is another popular way to express a linear equation. Here, \( m \) represents the slope and \( b \) is the y-intercept, the point where the line crosses the y-axis. Transitioning from the point-slope form to the slope-intercept form involves algebraic manipulation as shown in our exercise solution.

To illustrate, the original equation \( y-(-2)=\frac{3}{2}(x-1) \) simplifies to \( y=\frac{3}{2}x - \frac{7}{2} \) after distributing the slope and isolating \( y \). The result reveals that the line will cross the y-axis at \( -\frac{7}{2} \), a valuable piece of information when you're plotting the equation on a graph. This form is often the preferred method for quickly sketching a graph and understanding the line's behavior.

Plotting a linear equation on a graph visually represents the relationship between the x and y variables. Starting with the slope-intercept form, \( y=\frac{3}{2}x - \frac{7}{2} \), plot the y-intercept \( (0, -\frac{7}{2}) \) on the y-axis. As the slope is \( \frac{3}{2} \), which means 'rise over run,' you would go up 3 units and to the right 2 units from the y-intercept to find another point on the line.

Next, plot the point given in the point-slope form, \( (1,-2) \), to ensure consistency. This point should align with the line formed from using the slope. Drawing a straight line through these points completes the graph. Remember, the line represents all solutions to the equation; any point on the line is a solution. This graphical approach aids in visualizing the equation's impact and is a crucial skill in understanding linear relationships.

Interpreting slopes is a central aspect of understanding linear equations. The slope \( m \) in the equation indicates the line's steepness and the direction in which it tilts. A positive slope, such as \( \frac{3}{2} \) in our exercise, means that the line rises as it moves from left to right. Conversely, a negative slope indicates a falling line.

Slopes also reveal how much the \( y \) variable changes for a one-unit change in the \( x \) variable. In our equation's case, for every two units increase in \( x \) (the run), the \( y \) value rises by three units (the rise). Understanding slopes leads to better comprehension of the rate of change in related variables, a foundational concept for working with functions and modeling real-world scenarios.