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The point-slope form is one of the ways to express the equation of a straight line. It is particularly useful when you already know a point that lies on the line and the slope of the line itself. The general formula for point-slope is: \[y - y_1 = m(x - x_1)\] where:

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

By plugging in the values of the given point and the slope, you can quickly derive the equation of the line. It’s a starting point that can be simplified into other forms, including the slope-intercept form, if needed. In this specific exercise, using the point-slope formula immediately set us on the right path to finding the equation of the line by substituting \(m = 1\) and \((x_1, y_1) = (4, 7)\).

The slope is a crucial concept in understanding linear relationships. It determines the steepness and direction of a line on a graph. The slope \(m\) can be calculated if you have two points by using the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\] However, in this exercise, the slope is already provided as \(m = 1\).

• A positive slope like \(m = 1\) means the line inclines upwards as it moves from left to right.
• A negative slope would indicate a line that declines.
• A slope of zero means a horizontal line, and an undefined slope is a vertical line.

The value of the slope \(m = 1\) in this exercise informs us that for every unit increase in \(x\), the \(y\)-value increases by the same amount. Understanding the slope helps in visualizing the growth pattern or change rate within a dataset or equation.

Writing the equation of a line involves connecting both the slope and a point the line passes through. In various formats, these equations can represent different situations in coordinate geometry. From the point-slope format, the line's equation can be simplified or transformed, often into the slope-intercept form, which is: \[y = mx + b\] where \(b\) is the y-intercept, the point at which the line crosses the y-axis. In the original exercise, after using the point-slope form, we simplified and arrived at the slope-intercept form: \[y = x + 3\] Here:

• The slope \(m = 1\) reflects how steep the line is.
• The y-intercept \(b = 3\) indicates the line crosses the y-axis when \(x = 0\).

This form is very intuitive for graphing because it directly gives the starting point of the graph and the direction and rate of change. Different problems may start with different pieces of information, but understanding the equation in different forms allows flexibility in how to approach and solve them.