The table below gives some \((x, y)\) pairs that satisfy a linear relationship. What does \(z\) equal? $$\begin{array}{|c|c|}\hline x & {y} \\ \hline-2 & {-7} \\ \hline 2 & {5} \\\ \hline 0 & {-1} \\ \hline-3 & {z} \\ \hline\end{array}$$ E. -10 G. -8 H. -7 J. -2 K. 0

Understanding the slope of a line is crucial when dealing with linear relationships. Given two coordinate pairs, the slope represents how steep the line is and the direction it's going. It can tell you how much the 'y' value changes for a unit change in the 'x' value. To calculate the slope, also known as the rate of change, you use the formula:
\[\begin{equation} m = \frac{y_2 - y_1}{x_2 - x_1}\end{equation}\]
Where \begin{itemize}\item \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinate pairs,\item \( m \) is the slope of the line.\end{itemize}

The slope-intercept form of a line is perhaps one of the most friendly forms to work with in algebra. It is written as:
\[\begin{equation}y = mx + b\end{equation}\]
Where \begin{itemize}\item \( m \) is the slope of the line,\item \( b \) is the y-intercept, the point where the line crosses the y-axis.\end{itemize}

Coordinate pairs are the foundation of the coordinate plane system, and they come in the form of \((x, y)\). Each pair represents a point on a graph where 'x' indicates its horizontal position, and 'y' its vertical position. By plotting these points and connecting them, we can see the visual representation of mathematical relationships. For linear equations, two points are enough to determine the line.
In the exercise provided, we had three coordinate pairs and one incomplete pair. We used the complete pairs to establish the relationship, graphically the line, and then find the missing 'y' value (in our case, represented as \( z \)) for the specified 'x' value. This relationship allows us to predict or calculate unknown values such as \( z \), showing the power of linear equations in making inferences about data.