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Understanding how to calculate the slope is essential when working with linear functions. The slope of a line indicates how steep the line is. It is calculated by finding the change in the y-values divided by the change in the x-values between two distinct points. To calculate the slope, you can use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here, the points are given as

• \((x_1, y_1) = (2, -7)\)
• \((x_2, y_2) = (3, -5)\)

Substituting these into the formula, you get:\[ m = \frac{-5 - (-7)}{3 - 2} = \frac{-5 + 7}{1} = 2 \]This means that the slope of the line is 2. This tells us that for every 1 unit increase in the x-value, the y-value increases by 2 units. Grasping this concept makes understanding linear functions and their behaviors much easier.

The point-slope form is a powerful tool when working with linear equations. It helps you find the equation of a line if you know one point on the line and the slope.The general form is given by:\[ y - y_1 = m(x - x_1) \]This formula is great because you simply plug in the slope, and the coordinates of a point through which the line passes. Using this point-slope equation, you'll be able to form an easy-to-read equation of a line.In the example, we knew:

• The slope \((m) = 2\)
• Point \((x_1, y_1) = (2, -7)\)

Inserting these values into the point-slope formula results in:\[ y + 7 = 2(x - 2) \]This equation represents the line in its non-simplified form. It's a directly derived result of understanding the point-slope formula.

Simplifying the equation of a line makes it easier to interpret and use in further calculations, such as solving for specific values or graphing.From the point-slope equation \( y + 7 = 2(x - 2) \), we simplify it to get a more standard linear form, known as the slope-intercept form, \( y = mx + b \).Follow these steps to simplify:1. Distribute the slope across the terms in the parentheses: \( 2(x - 2) = 2x - 4 \)2. You then have \( y + 7 = 2x - 4 \). Next, we want to isolate \(y\) by subtracting 7 from both sides: \[ y = 2x - 4 - 7 \]3. This ultimately simplifies to: \[ y = 2x - 11 \]The final result, \( y = 2x - 11 \), is in the slope-intercept form, indicating a slope of 2 and a y-intercept of -11, making it extremely useful for graphing or analyzing the behavior of this line. Equation simplification allows for easier manipulation and understanding of linear equations.