91Ó°ÊÓ

The slope-intercept form of a linear equation is probably the most common and easily recognizable form. It is expressed as:

• \( y = mx + c \)

Here, \(m\) represents the slope of the line, and \(c\) is the y-intercept, where the line crosses the y-axis. This form is crucial because the slope \(m\) gives us a direct understanding of the angle at which the line tilts. If the slope is positive, the line rises; if negative, it falls. The y-intercept \(c\) helps us find the starting point of the line on the y-axis.

In the existing solution, the line equation \(3y - 2x + 6 = 0\) was rearranged into slope-intercept form \(y = \frac{2}{3}x - 2\), revealing a slope \(m = \frac{2}{3}\). Knowing this slope is key for finding equations of parallel lines since parallel lines share the same slope.

Sometimes, when constructing the equation of a line, you already know a specific point on the line and the slope. The point-slope form of a linear equation is tailor-made for these scenarios, expressed as:

• \( y - y_1 = m(x - x_1) \)

In this form, \((x_1, y_1)\) represents a known point on the line, and \(m\) denotes the slope of the line. This makes it incredibly useful when you have a point and a slope and need to write the equation of a line.

For the current exercise, since the line must be parallel to the given line, it uses the same slope \(m = \frac{2}{3}\). Given the point \(P(2,7)\), we plug these into the point-slope form to get \(y - 7 = \frac{2}{3}(x - 2)\). After simplifying, this can be rewritten in slope-intercept form as \(y = \frac{2}{3}x + \frac{5}{3}\).

The standard form of a line is another way of presenting a linear equation, generally noted as:

• \( Ax + By = C \)

In standard form, \(A\), \(B\), and \(C\) are integers, and \(A\) should ideally be positive. It is beneficial when needing a clean integer representation of the line, which can be useful for various types of analyses and graphing.

Converting from slope-intercept form to standard form becomes straightforward by eliminating any fractions through multiplying by the least common multiple. In the solution, the slope-intercept form \(y = \frac{2}{3}x + \frac{5}{3}\) was multiplied by 3 to clear the fractions, leading to the standard form: \(2x - 3y = -5\). This transformation gives a clear, fraction-free equation, commonly preferred for its simplicity and utility in different mathematical applications.