91Ó°ÊÓ

The slope of a line can be found using the formula \(m=\frac{y_2-y_1}{x_2-x_1}\), where \(m\) is the slope, and \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points on the line. Using the given points \((3,5)\) and \((-1,2)\), let's find the slope: \(m = \frac{2 - 5}{-1 - 3} = \frac{-3}{-4} = \frac{3}{4}\)

The point-slope form of a line equation is given by \(y - y_1 = m(x - x_1)\), where \(m\) is the slope, and \((x_1, y_1)\) is any point on the line. Using the slope and one of the given points (e.g., \((3,5)\)), we can find the equation of the line: \(y - 5 = \frac{3}{4}(x - 3)\)

To get the equation of the line in slope-intercept form, solve for \(y\). To do this, multiply both sides of the equation by the denominator of the slope (4): \(4(y - 5) = 4 \left( \frac{3}{4}(x - 3) \right)\) This simplifies to: \(4y - 20 = 3(x - 3)\) Now, distribute the 3 into the expression inside the parenthesis: \(4y - 20 = 3x - 9\) Next, move all the terms involving \(y\) to the left side and all the terms involving \(x\) to the right side: \(4y = 3x - 9 + 20\) Now, simplify the right side and divide both sides by 4 to get the equation in slope-intercept form: \(y = \frac{3}{4}x + \frac{11}{4}\) So, the equation of the line passing through \((3,5)\) and \((-1,2)\) is \(y = \frac{3}{4}x + \frac{11}{4}\).