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Calculating the slope of a line is a fundamental skill in understanding how lines work on a graph. The slope is expressed as the ratio of the change in the vertical direction (y-axis) to the change in the horizontal direction (x-axis). This is a measure of how steep the line is. In mathematical terms, the formula for finding the slope \( m \) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]To use this formula, you need two points on the line. For example, with points \((\frac{1}{2}, 1)\) and \((2, 7)\), substitute them into the formula:- Start by calculating the difference in the \( y \)-values: \( 7 - 1 = 6 \)- Then, calculate the difference in the \( x \)-values: \( 2 - \frac{1}{2} = \frac{3}{2} \)- Now divide the differences: \( \frac{6}{\frac{3}{2}} = 4 \)So, the slope \( m \) of the line is 4. Each unit of increase in \( x \) corresponds to a 4-unit increase in \( y \). This concept of slope is essential when graphing lines and understanding how they relate to their equations.

The point-slope form of a linear equation is a straightforward way to express the equation of a line when you know the slope and one point on the line. The general formula is given by:\[ y - y_1 = m(x - x_1) \]Here, \((x_1, y_1)\) is a known point on the line, and \( m \) is the slope. Starting with the points \((\frac{1}{2}, 1)\) and the slope \( m = 4 \) calculated previously, use the point-slope formula:- Substitute the values into the formula: \( y - 1 = 4(x - \frac{1}{2}) \)This equation states that for the line with a slope of 4, any point \((x, y)\) on the line will satisfy this equality. This form is useful because it immediately highlights both the slope and a specific point used to derive the equation. It is often a helpful starting place before converting into other forms of linear equations.

A linear equation describes a straight line on a graph, commonly written in slope-intercept form \( y = mx + b \). Here:- \( y \) is the dependent variable,- \( x \) is the independent variable,- \( m \) is the slope, and- \( b \) is the y-intercept, the point where the line crosses the y-axis.From the point-slope equation \( y - 1 = 4(x - \frac{1}{2}) \), the goal is to write it in slope-intercept form:- Distribute the slope across the point-term: \( y - 1 = 4x - 2 \)- Isolate \( y \) by adding 1 to both sides: \( y = 4x - 2 + 1 \)- Simplify to get \( y = 4x - 1 \)The equation \( y = 4x - 1 \) succinctly represents the line’s characteristics. The slope \( m \) is 4, indicating steepness, and the y-intercept \( b = -1 \) shows where the line crosses the y-axis. Understanding this form allows for quick graphing and interpretation of linear relationships. It is one of the most widely-used representations for lines in algebra.