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Calculating the slope of a linear equation is an essential skill in algebra, as it describes how steep a line is. The slope is typically represented by the letter \(m\) and is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line. The numerator \((y_2 - y_1)\) measures the change in the vertical direction, also known as "rise," while the denominator \((x_2 - x_1)\) measures the change in the horizontal direction, or "run."

• When \(m > 0\), the line slopes upward, indicating a positive correlation.
• When \(m < 0\), the line slopes downward, showing a negative correlation.
• A slope of \(0\) means the line is perfectly horizontal.

Understanding the concept of slope allows you to determine how a change in \(x\) affects \(y\). For example, in the given problem of connecting the points \((-1, 4)\) and \((5, 2)\), the slope was calculated as \(-\frac{1}{3}\), indicating a downward-sloping line.

The point-slope form is a powerful tool for writing the equation of a line when you know one point on the line and its slope. This form is given by the formula: \[ y - y_1 = m(x - x_1) \] where \((x_1, y_1)\) is a known point on the line and \(m\) is the slope.
Using the point-slope form, you can easily write the equation of a line. You need:

• The slope \(m\), which we've already found as \(-\frac{1}{3}\).
• The coordinates of a point on the line, such as \((-1, 4)\).

Substituting these values into the point-slope formula gives:\[ y - 4 = -\frac{1}{3}(x + 1) \] This expresses the relationship between \(x\) and \(y\) for any point on the line. It's beneficial because it directly uses the slope and a specific point, making it clear and straightforward to apply.

The slope-intercept form is a specific linear equation expression that makes it easy to identify a line's slope and y-intercept at a glance. The general form of the slope-intercept equation is: \[ y = mx + b \] where \(m\) is the slope and \(b\) is the y-intercept.
The y-intercept \(b\) represents the point where the line crosses the y-axis, which is when \(x = 0\). To convert from point-slope form to slope-intercept form, simply distribute and simplify. Following this method for our equation: \[ y - 4 = -\frac{1}{3}(x + 1) \] leads to:

• Distributing: \(y - 4 = -\frac{1}{3}x - \frac{1}{3}\)
• Adding 4 to both sides to solve for \(y\): \(y = -\frac{1}{3}x + \frac{11}{3}\)

Now, you can easily see the line has a slope of \(-\frac{1}{3}\) and crosses the y-axis at \(\frac{11}{3}\). Knowing the slope-intercept form is incredibly useful in graphing and understanding linear relationships quickly.