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The slope of a line represents how steep the line is. It is calculated using two coordinate points, designated as \((x_1, y_1)\) and \((x_2, y_2)\). The formula for finding the slope is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \],where \(m\) is the slope. For a practical example, let's use the points \((3, 4)\) and \((-3, -10)\). By substituting these values into the slope formula, we calculate:

• \( y_2 - y_1 = -10 - 4 = -14 \)
• \( x_2 - x_1 = -3 - 3 = -6 \)

Thus, the slope is calculated as \[ m = \frac{-14}{-6} = \frac{7}{3} \].Breaking it down, the slope of \( \frac{7}{3} \) indicates that for every increase of 3 units in \(x\), \(y\) increases by 7 units.

The y-intercept is a critical part of the slope-intercept form equation, \(y = mx + b\). It represents the point where the line crosses the y-axis and can be found by setting \(x\) to 0.To find the y-intercept, \(b\), use the slope \(m\) and one known point from the line. For example, using the calculated slope \(m = \frac{7}{3}\) and the point \((3, 4)\), insert these values into the slope-intercept formula:\[ 4 = \frac{7}{3}(3) + b \].Calculate \( \frac{7}{3} \times 3 = 7 \) and substitute back to find:\[ 4 = 7 + b \].Subtract 7 from both sides to solve for \(b\):\[ b = -3 \].Thus, the y-intercept is -3, indicating that our line crosses the y-axis at the point \((0, -3)\).

Linear equations describe a line on a 2-dimensional plane and come in different forms. Here, we focus on the slope-intercept form, noted as: \[ y = mx + b \].This structure is beneficial because it provides a straightforward way to understand a line's behavior through \(m\), the slope, and \(b\), the y-intercept.

• The slope \(m\) tells us the steepness and direction (positive slope = upward, negative = downward).
• The y-intercept \(b\) tells us where the line crosses the y-axis.

In our example, the linear equation derived is \[ y = \frac{7}{3}x - 3 \].This means that starting at \((0, -3)\) on the y-axis, every move right by 3 units results in a rise of 7 units, defining the line's graph.

Coordinate points are fundamental in plotting and understanding lines. They denote specific locations on the Cartesian plane, given in the form \((x, y)\). Each point on a line can be used to derive other points or properties such as the slope and y-intercept.For the given points \((3, 4)\) and \((-3, -10)\), these coordinates act as specific solutions to the linear equation they define. By comparing these two:

• \( (3, 4) \) tells you that when \(x\) is 3, \(y\) is 4.
• \( (-3,-10) \) shows that when \(x\) is -3, \(y\) is -10.

These points are used in the slope formula to calculate the line's incline and are plugged into the slope-intercept formula to find the y-intercept. Understanding coordinate points makes interpreting and graphing lines possible, enabling us to visualize how mathematics applies to the world around us.