91Ó°ÊÓ

The slope of a line, represented as \(m\), measures the steepness and direction of the line. It is crucial for forming the line equation. To find the slope between two points, you use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] - **Step-by-step:** You subtract the y-coordinates of the two points from each other and divide by the difference of their x-coordinates. - **Example Calculation:** For points \(\left(\frac{3}{5}, \frac{4}{10}\right)\) and \(\left(-\frac{1}{5}, \frac{7}{10}\right)\), this is: \[ m = \frac{\frac{7}{10} - \frac{4}{10}}{-\frac{1}{5} - \frac{3}{5}} = \frac{\frac{3}{10}}{-\frac{4}{5}} = -\frac{3}{8} \] It is important to accurately perform arithmetic operations involving fractions to determine the correct slope.

Once you have calculated the slope, you can use the point-slope form of a linear equation. This form is useful for writing the equation of a line with a known slope that passes through a specific point. The formula is: \[ y - y_1 = m(x - x_1) \] - **Explanation:** This equation is straightforward. You insert the slope \(m\) and a point \((x_1, y_1)\) on the line. - **Practical Use:** For our example, using slope \(-\frac{3}{8}\) and point \(\left(\frac{3}{5}, \frac{4}{10}\right)\), plug these values in as follows: \[ y - \frac{4}{10} = -\frac{3}{8}(x - \frac{3}{5}) \] This equation can then be further rearranged to simplify.

To translate from the point-slope form to a simplified line equation, which is often needed for various applications, you simplify and rearrange the expression into either slope-intercept or standard form. Simplifying the equation helps in understanding the relationship between x and y. - **Process:** From the point-slope we previously formed, simplify each side and clear fractions: \[ y - \frac{2}{5} = -\frac{3}{8}x + \frac{9}{40} \] By rearranging and simplifying further, errors are avoided, especially with fractions. This will form a clear association between x and y coordinates.

Standard form is typically expressed with integer coefficients, i.e., \(Ax + By = C\), without fractions or decimals. Ensuring integer coefficients is essential for the equation to be in 'standard form,' which is preferred for mathematical applications. - **Steps to Achieve:** Multiply every term by a common denominator to clear all fractions. Then, arrange terms so \(A\) is positive. - **Example Transformation:** For line equation: \[ y = -\frac{3}{8}x + \frac{25}{40} \] Multiply by 40: \[ 40y = -15x + 25 \] - **Final Rearrangement:** Swap terms and ensure \(A\) is positive: \[15x + 40y = 25\] This achieves the required format with integer coefficients.