91Ó°ÊÓ

Understanding how to graph a linear equation begins with a simple yet powerful tool: the table of values. This method involves choosing specific values for one variable and then calculating the corresponding values for the other variable. In the case of the equation \( y = 3x - 7 \), we first select several values for \( x \). For instance, values like -3, -1, 0, 1, and 3 are good choices because they are integers and make the calculations easier.

For each chosen \( x \)-value, we substitute it into our equation and solve for \( y \). We record both \( x \) and \( y \) in the table to obtain coordinate points, which will be used to plot on the graph. This process not only helps in visualizing the relationship between \( x \) and \( y \) but also provides a clear step-by-step approach to graphing. A well-constructed table of values is like a roadmap that guides you smoothly to the final graph.

Coordinate points are the next stepping stones in graphing our linear equation once the table of values is ready. Each pair of \( x \) and \( y \) values from our table correlates to a single point on the coordinate plane, consisting of a horizontal x-axis and a vertical y-axis.

To plot a point such as \( (-3, -16) \), locate the \( x \)-value on the horizontal axis and the \( y \)-value on the vertical axis. Where these values intersect is the exact point you'll mark on the graph. The coordinate point represents a solution to the equation, and when multiple points are plotted, they reveal a pattern. Connecting these dots will lead us to the visual representation of our equation—a straight line. It's essential to be precise when plotting points to ensure that the resulting graph accurately reflects the equation.

The slope of a line is a measure of its steepness and direction. It's a vital concept in understanding linear equations as it describes how much \( y \) changes for a change in \( x \). For the given equation \( y = 3x - 7 \), the number 3, which multiplies the \( x \)-variable, is the slope. This tells us that for each increase by 1 in \( x \), \( y \) increases by 3 units.

The slope being positive means our line rises from left to right. When drawing this line on the graph, you can visualize this by noting that the line moves upward as you follow it from left to right. If you're familiar with the slope formula \( m = (y_2 - y_1) / (x_2 - x_1) \), the slope can also be calculated using any two coordinate points from the line. Just remember, the consistent upward or downward movement of a line across a graph is directly linked to the sign and value of its slope.