Problem 46 - College Algebra

Question

Give the slope and y-intercept of each line whose equation is given. Then graph the line. $$y=-\frac{2}{5} x+6$$

Step-by-Step Solution

Answer

The slope of the line is -\frac{2}{5} and the y-intercept is 6. The line can be graphed using these values, starting at the point (0,6) and then moving down 2 units and to the right 5 units for the next point.

Step 1: Identify the Slope

For the given equation \(y = -\frac{2}{5}x + 6\), the coefficient of \(x\) is \(m = -\frac{2}{5}\). This is the slope of the line.

Step 2: Identify the y-intercept

In the same equation, \(b = 6\) is the y-intercept. This is the point where the line crosses the y-axis.

Step 3: Graph the Line

Start by marking the y-intercept (\(0,6\)) on the graph. From this point, use the slope to find additional points and draw the line. Remember, slope is defined as rise over run, so the slope of -\frac{2}{5} means drop down 2 units and move right 5 units from the y-intercept.

Key Concepts

Slope

The slope of a line is a measure of its steepness or tilt. In the slope-intercept form of a linear equation, which is given by \(y = mx + b\), the slope is represented by the coefficient \(m\). It tells you how much the \(y\) value changes with a change in the \(x\) value. A positive slope indicates the line rises as it moves from left to right, while a negative slope means the line falls. For example, in the equation \(y = -\frac{2}{5}x + 6\), the slope is \(-\frac{2}{5}\). This negative slope suggests that as you move 5 units to the right, the line will drop 2 units. Understanding slope is crucial for graphing lines as it dictates the direction and angle of the line on the graph.

y-intercept

The y-intercept is the point where the line crosses the y-axis. It is a key feature of linear equations in the slope-intercept form. In the equation \(y = mx + b\), the \(b\) represents the y-intercept. This is the point \((0, b)\) on the graph, indicating where the line will intersect the y-axis. In our given equation \(y = -\frac{2}{5}x + 6\), the y-intercept is 6. This tells us that when \(x = 0\), \(y\) will equal 6. The y-intercept provides a starting point for graphing the line and, combined with the slope, allows us to plot the entire graph accurately.

Graphing Linear Equations

Graphing linear equations is about visually representing the relationship between \(x\) and \(y\). To graph the equation \(y = -\frac{2}{5}x + 6\), start by plotting the y-intercept, which is the point \((0, 6)\) on the y-axis. From this point, use the slope to find additional coordinates. With the slope \(-\frac{2}{5}\), follow these steps:

• From the y-intercept, move down 2 units (because the numerator of the slope is -2, indicating a downward direction).
• Then, move right 5 units (as per the positive denominator of the slope).

Place another point at this new location. Draw a straight line through these points, extending it across the graph. This line represents the equation and shows all solutions \((x, y)\) that satisfy it. By understanding the slope and the y-intercept, graphing becomes a straightforward process.