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To calculate the slope of a line, which is often denoted by the letter \(m\), we use a straightforward formula. The slope is defined as the change in the \(y\)-coordinates divided by the change in the \(x\)-coordinates between two distinct points on a line. Mathematically, this is represented as: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula gives us an idea of how steep the line is. If \(m\) is positive, the line rises as we move along the \(x\)-axis; if it's negative, the line falls as we move.

For our specific example with points \((0,-2)\) and \((-6,1)\), we apply the formula as follows:

• Identify your points: \((x_1, y_1) = (0, -2)\) and \((x_2, y_2) = (-6, 1)\)
• Calculate the differences: \(y_2 - y_1 = 1 - (-2) = 3\) and \(x_2 - x_1 = -6 - 0 = -6\)
• Formulate the slope: \(m = \frac{3}{-6} = -\frac{1}{2}\)

This tells us that for every two units we move horizontally to the right, the line goes down by one unit vertically.

The \(y\)-intercept of a line, represented by \(b\), is the point where the line crosses the \(y\)-axis. This is the point at \(x = 0\), where the line intersects the vertical axis, indicating the initial value of \(y\) when there are no \(x\) inputs.

To find the \(y\)-intercept, we use the point-slope form of a line and substitute one of the points along with the calculated slope. Using the slope-intercept form \(y = mx + b\), choose a point, say \((0, -2)\).

• Substitute \(m = -\frac{1}{2}\) and the point \((x, y) = (0, -2)\):
• \(-2 = -\frac{1}{2} \times 0 + b\)
• Solving for \(b\), we find \(b = -2\)

Thus, the line intersects the \(y\)-axis at \(y = -2\). This means the line starts from \(-2\) on the \(y\)-axis when \(x = 0\).

Once both the slope \(m\) and the \(y\)-intercept \(b\) are known, the line's equation can be constructed in the slope-intercept form, which is very user-friendly and easy to interpret. This form is expressed as:\[y = mx + b\]It provides a direct representation of the line where \(m\) is the slope, showing how the line moves, and \(b\) indicates where the line crosses the \(y\)-axis.

From the exercise, we've calculated the slope as \(-\frac{1}{2}\) and the \(y\)-intercept as \(-2\). By substituting these values into the slope-intercept formula, we get:\[y = -\frac{1}{2}x - 2\]This equation tells us exactly what our line looks like on a graph:

• For every step to the right (increase in \(x\)), we move half a step downwards (decrease in \(y\)).
• The line starts intersecting the \(y\)-axis at \(-2\).

This form of equation is simple and especially helpful in graphing or interpreting the line's behavior in different situations.