What does the slope represent in the equation y = x/2 - 1?

The slope represents the rate of change of y with respect to x, which is 1/2 in this equation.

In the equation \( y = \frac{x}{2} - 1 \), the slope is the coefficient of \( x \), which is \( \frac{1}{2} \). This means that for every unit increase in \( x \), \( y \) increases by \( \frac{1}{2} \) units. The slope tells us how steep the line is and the direction it goes. A positive slope, like \( \frac{1}{2} \), indicates that the line rises as it moves from left to right.

To understand this better, imagine plotting the equation on a graph. Start by choosing a value for \( x \), say \( x = 0 \). Plugging this into the equation gives \( y = \frac{0}{2} - 1 = -1 \). So, one point on the graph is (0, -1). Now, choose another value for \( x \), such as \( x = 2 \). Plugging this in gives \( y = \frac{2}{2} - 1 = 0 \). So, another point is (2, 0). If you plot these points and draw a line through them, you'll see that the line rises by 1 unit for every 2 units it moves to the right, which matches our slope of \( \frac{1}{2} \).

The slope is crucial in understanding how the line behaves. If the slope were negative, the line would fall as it moves to the right. If the slope were larger, the line would be steeper. In summary, the slope \( \frac{1}{2} \) in this equation tells us that the line rises gently, increasing by half a unit in \( y \) for each unit increase in \( x \).