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The point-slope form is a simple yet powerful way to describe the equation of a line when you know a point on the line and the slope. The formula is: \( y - y_1 = m(x - x_1) \), where:

• \( (x_1, y_1) \) is a known point on the line.
• \( m \) is the slope of the line.

This form is particularly helpful because you can plug in the values directly and quickly derive the equation of the line. It serves as a bridge to convert real-world situations into mathematical equations. For instance, if you are given a slope of \( \frac{3}{2} \) and a point \( (2, 1) \), you replace them in the formula to get \( y - 1 = \frac{3}{2}(x - 2) \), which reveals the line's equation after simplification.

Parallel lines share the same slope. This means that if you can express their equations in the form \( y = mx + c \), both lines will have the same \( m \) value. This property is useful when you need to find a line that is parallel to a given line. For example, consider the line represented by \( 3y - x = 5 \). By rearranging it to slope-intercept form, \( y = \frac{1}{3}x + \frac{5}{3} \), you find the slope is \( \frac{1}{3} \). Thus, any line parallel to this must also have a slope of \( \frac{1}{3} \). To find a parallel line passing through \( (1, 1) \), just use the point-slope form with these values, resulting in \( y = \frac{1}{3}x + \frac{2}{3} \) after simplification.

Perpendicular lines have slopes that are negative reciprocals of each other. This means that if the slope of one line is \( m \), the slope of a line perpendicular to it will be \( -\frac{1}{m} \). This relationship is crucial in many geometric and algebraic contexts where perpendicularity is required.Taking the earlier line \( 3y - x = 5 \) as an example, its slope is \( \frac{1}{3} \). Therefore, the slope of a line perpendicular to it would be \( -3 \), since \( -\frac{1}{\frac{1}{3}} = -3 \). If you want to find the line that is perpendicular to this and passes through \( (1, 1) \), use the point-slope form to derive \( y = -3x + 4 \).

The slope of a line is a measure of its steepness and direction. To compute the slope \( m \) between any two points \( (x_1, y_1) \) and \( (x_2, y_2) \), use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This process involves calculating the "rise" over the "run, " which is simply the change in \( y \) divided by the change in \( x \). Such calculation is fundamental for translating points into linear equations, or vice versa.For instance, to find the slope between points \( (1, 2) \) and \( (3, 7) \), plug into the formula: \( m = \frac{7 - 2}{3 - 1} = \frac{5}{2} \). This tells you how much the line steps up for each unit it moves to the right. Knowing the slope allows you to proceed with using the point-slope form to find the line's exact equation.