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When we talk about the point-slope form of a linear equation, we're referring to a specific format used to define a straight line. This is especially useful when we have a point on the line and the slope of the line. The point-slope equation is:\[ y - y_1 = m(x - x_1) \]Where:

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

In the exercise provided, we transformed our equation by inserting the negative reciprocal slope of \(-\frac{5}{2}\), which is perpendicular to the slope \(\frac{2}{5}\) of the given line, and the point \( A(7, -3) \). This allows us to produce an equation in point-slope form:\[ y + 3 = -\frac{5}{2}(x - 7) \]This form is convenient for many applications because it directly uses a point on the line, reflecting situations where you don't need to compute additional intercepts as you do in other forms.

The slope-intercept form is another way of expressing the equation of a line. It's popular in algebra because it provides immediate insight into the slope and y-intercept of the line. The slope-intercept form is:\[ y = mx + b \]Here, \( m \) represents the slope, and \( b \) is the y-intercept where the line crosses the y-axis. To transform from point-slope to slope-intercept form:

• First, solve for \( y \) in terms of \( x \).
• Simplify the resulting expression.

In the exercise, the given line was rewritten in this form to identify the slope:\[ y = \frac{2}{5}x - \frac{8}{5} \]By transforming to slope-intercept form, it becomes evident that the slope is \(\frac{2}{5}\) and the y-intercept is \(-\frac{8}{5}\). For the line we created, direct conversion isn't necessary, but understanding this format helps in visualizing and sketching graphs.

The general form of a line equation is another commonly used representation. This standard format is expressed as:\[ Ax + By + C = 0 \]Where \( A, B, \) and \( C \) are integers, and typically \( A \) should be a non-negative integer. The advantage of this form is its unified representation of all linear equations, even vertical lines, which cannot be expressed in slope-intercept form.For the exercise, after using point-slope form, we converted the equation to the general form with these steps:

• Begin with the point-slope equation.
• Clear fractions and rearrange terms.

This process led us to the general form:\[ 5x + 2y - 29 = 0 \]This form is useful for analyzing and comparing different lines, as it provides a straightforward way to discern intersection points and parallelism by examining the coefficients.