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Understanding the slope-intercept form is a key step in graphing linear equations. This form of an equation for a straight line is expressed as: \[ y = mx + b \] Where: * \(m\) represents the slope of the line. * \(b\) represents the \(y\)-intercept of the line. The line will cross the y-axis at the point \(b\). Knowing the slope-intercept form makes it easier to quickly identify and graph the slope and the y-intercept when given. For example, to graph the line with the equation \[ y = -\frac{1}{3}x + 5 \], we recognize that the slope (\(m\)) is \(-\frac{1}{3}\) and the y-intercept (\(b\)) is 5.

The slope of a line is a measure of its steepness and direction. It's calculated as the ratio of the vertical change to the horizontal change between two points on the line. This is often expressed as: \[ m = \frac{\Delta y}{\Delta x} \] Here, \(\Delta y\) represents the change in y-values, while \(\Delta x\) represents the change in x-values. If the slope (\(m\)) is negative, the line will slope downwards from left to right. In our example, the slope -\(\frac{1}{3}\) means that for every 3 units we move to the right (positive x-direction), we move 1 unit down (negative y-direction). Visualizing this can help in plotting points accurately on the graph for a given slope.

Plotting points is one of the fundamental steps in graphing linear equations. First, you start with the y-intercept, which is where the line crosses the y-axis. For the given exercise: * Start by plotting the y-intercept (0, 5) on the graph. * Next, use the slope to determine another point. With a slope of -\(\frac{1}{3}\), you begin at (0, 5), move 3 units to the right, and then 1 unit down. This gets you to the point (3, 4). * Mark this point on the graph. After plotting these points, draw a straight line through them. This line represents your linear equation y = -\(\frac{1}{3}\)x + 5. Plotting points correctly is crucial for accurately representing the equation on the graph.