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The slope-intercept form is one of the most useful ways to express the equation of a line. This format makes it easy to identify both the slope and the y-intercept of a line. The standard slope-intercept equation is written as:

• \( y = mx + b \)
• where \( m \) represents the slope of the line
• and \( b \) represents the y-intercept, the point where the line crosses the y-axis

Knowing the slope is crucial when dealing with parallel lines. For two lines to be parallel, their slopes must be identical. This means if a line in slope-intercept form has a slope of \( \frac{2}{3} \), then any line parallel to it will also have a slope of \( \frac{2}{3} \). Additionally, the intercept \( b \) can vary between parallel lines, allowing them to be distinct lines that never meet.
Looking at our exercise, the given line is \( y = \frac{2}{3}x - 7 \), readily showing that the slope \( m = \frac{2}{3} \).

The point-slope form is another invaluable tool for expressing the equation of a line. This form is particularly useful when you have a point and the slope. Here's the equation for the point-slope form:

• \( y - y_1 = m(x - x_1) \)
• Where \((x_1, y_1)\) is a specific point on the line
• And \( m \) is the slope

This format is extremely handy when you know a single point and need to find the equation of the line that runs through it. By plugging in the slope and coordinates of the point, you can derive the full equation. This form transitions nicely into slope-intercept or standard form for additional clarity.
In our situation, with a slope \( m = \frac{2}{3} \) and the point \((6, 0)\), we use the point-slope form as:\[ y - 0 = \frac{2}{3}(x - 6) \].

Constructing the equation of a line entails placing it into one of several common forms, most generally known through the slope-intercept or point-slope forms. Each form serves a unique purpose depending on the data you have. In our exercise, we've effectively utilized both forms to derive a specific equation.To change from point-slope to slope-intercept, you simply solve for \( y \). In our exercise scenario, after substituting into the point-slope form, we simplified it as:

• Start: \( y - 0 = \frac{2}{3}(x - 6) \)
• Distribute \( \frac{2}{3} \): \( y = \frac{2}{3}x - \frac{12}{3} \)
• Final simplified form: \( y = \frac{2}{3}x - 4 \)

This final equation \( y = \frac{2}{3}x - 4 \) represents a line parallel to the original one \( y = \frac{2}{3}x - 7 \) but shifted vertically to pass through the point \((6, 0)\). Understanding these classical forms allows you to jaunt seamlessly between them, showcasing the versatility of line equations.