For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. See Examples 3 and 4. y = ½ x - 2

Verified step by step guidance

1. Identify the given equation: \(y = \frac{1}{2}x - 2\). This is a linear equation in slope-intercept form \(y = mx + b\), where \(m = \frac{1}{2}\) is the slope and \(b = -2\) is the y-intercept.
2. To create a table of ordered pairs \((x, y)\), choose at least three values for \(x\). For each chosen \(x\), substitute it into the equation to find the corresponding \(y\) value.
3. For example, select \(x = 0\), \(x = 2\), and \(x = 4\). Calculate \(y\) for each: - When \(x=0\), \(y = \frac{1}{2} \times 0 - 2 = -2\). - When \(x=2\), \(y = \frac{1}{2} \times 2 - 2 = 1 - 2 = -1\). - When \(x=4\), \(y = \frac{1}{2} \times 4 - 2 = 2 - 2 = 0\).
4. Organize these results into a table of ordered pairs: \[\begin{array}{c|c}$ x & y \(\hline\) 0 & -2 \\ 2 & -1 \\ 4 & 0 $\end{array}\]
5. To graph the equation, plot the points from the table on the coordinate plane. Then, draw a straight line through these points, extending it in both directions. The slope \(\frac{1}{2}\) means the line rises 1 unit for every 2 units it moves to the right.

Key Concepts

Slope-Intercept Form of a Linear Equation

The equation y = ½ x - 2 is in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. The slope ½ indicates the line rises 1 unit for every 2 units it moves right, and the y-intercept -2 is where the line crosses the y-axis.

Finding Ordered Pairs (Solutions) for a Linear Equation

To find ordered pairs (x, y) that satisfy the equation, select values for x and compute the corresponding y values using the equation. These pairs represent points on the line and are essential for plotting the graph.

Graphing a Linear Equation

Graphing involves plotting the ordered pairs on a coordinate plane and drawing a straight line through them. The slope determines the line's angle, and the y-intercept shows where it crosses the y-axis, providing a visual representation of all solutions.