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When finding the equation of a line, the first key concept is the slope. The slope measures how steep the line is and the direction it tilts. We calculate slope using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula uses the coordinates of two points on the line. In the given problem, the points are \( \left( \frac{3}{4}, \frac{8}{3} \right) \) and \( \left( \frac{2}{5}, \frac{2}{3} \right) \). By substituting these values into the formula, we get: \( m = \frac{\frac{2}{3} - \frac{8}{3}}{\frac{2}{5} - \frac{3}{4}} = \frac{-2}{-\frac{7}{20}} = \frac{40}{7} \). Note that the negative signs cancel each other out, giving us the slope \( \frac{40}{7} \). This slope tells us how much the line rises vertically for every unit it moves horizontally.

After finding the slope, the next step is to use the point-slope form of the equation of a line. The point-slope form is expressed as: \( y - y_1 = m(x - x_1) \). It is particularly useful as it easily incorporates both a point on the line and the slope. Using our slope \( m = \frac{40}{7} \) and one of the points \( \left( \frac{3}{4}, \frac{8}{3} \right) \), we plug these values into the formula: \( y - \frac{8}{3} = \frac{40}{7} (x - \frac{3}{4}) \). This rearranges our line equation based on a specific point and the calculated slope, forming the foundation to convert to other line equation forms.

The slope-intercept form is often the most intuitive and commonly used form for a line equation. It is written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. From the point-slope form \( y - \frac{8}{3} = \frac{40}{7} (x - \frac{3}{4}) \), we distribute and simplify to find \( b \). First, distribute the slope: \( y - \frac{8}{3} = \frac{40}{7} x - \frac{30}{7} \). Then add \( \frac{8}{3} \) to both sides. Finding a common denominator, we get: \( y = \frac{40}{7} x - \frac{30}{7} + \frac{56}{21} \). Simplifying gives: \( y = \frac{40}{7} x - \frac{2}{21} \). Here, \( \frac{40}{7} \) is the slope and \( -\frac{2}{21} \) is the y-intercept.

The standard form of a line equation is expressed as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should be positive. To convert the slope-intercept form \( y = \frac{40}{7} x - \frac{2}{21} \) to standard form, we first eliminate the fractions by multiplying through by 21: \( 21y = 120x - 2 \). To align it into the standard structure, we rearrange terms to get \( 120x - 21y = 2 \). This equation now meets the standard form criteria.