Problem 40 - College Algebra and Calculus: An Applied Approach

Question

Find the slope and \(y\) -intercept (if possible) of the line specified by the equation. Then sketch the line. \(y=3-x\)

Step-by-Step Solution

Answer

The slope of the line is -1 and the y-intercept is 3.

Step 1: Identify the slope and y-intercept

The equation \(y=3-x\) is in the form \(y = -x + 3\); a standard form of line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.\nSo, comparing these two, it can be determined that the slope \(m = -1\) and the y-intercept \(b = 3\).

Step 2: Plot the y-intercept

The y-intercept is the point where the line crosses the y-axis. Here, the y-intercept is \(b = 3\), which corresponds to the point (0,3). Start there and place a dot.

Step 3: Use the slope to find a second point and draw the line

The slope is the rise over run. Here, the slope is \(m = -1\) which can be written as \(-1/1\), meaning down 1 unit and right 1 unit from the y-intercept. Mark that spot and then draw a line through the two points.

Key Concepts

Slope

The slope of a line is a measure of its steepness and is represented by the letter \(m\) in the slope-intercept form, which is \(y = mx + b\). In this equation, \(m\) denotes how tilted or inclined the line is. To find the slope, look at the number multiplied by \(x\). If there's no visible number, like in the equation \(y = 3 - x\), it's actually \(-1\), since the equation can be rearranged to \(y = -x + 3\). The slope tells you the following:

• Positive slope (\(m > 0\)): The line rises as you move from left to right.
• Negative slope (\(m < 0\)): The line falls as you move from left to right, like \(m = -1\) in our example.
• Zero slope (\(m = 0\)): The line is horizontal, indicating no rise or fall.
• Undefined slope: This occurs with vertical lines where the change in \(x\) is zero.

For \(y = 3 - x\), the slope \(m = -1\), tells us that for every step we move to the right along the x-axis, we move one step downwards on the y-axis.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form \(y = mx + b\), the y-intercept is represented by \(b\). It tells us the specific point where the line will meet or intersect the y-axis. Knowing the y-intercept helps us place one of the critical points when sketching the graph.For our line described by \(y = 3 - x\), the y-intercept is \(b = 3\). This means the line crosses the y-axis at the point (0, 3). This point is crucial as a starting point when plotting the graph of the line. In practical terms, you can think of the y-intercept as the starting value or initial condition of the line when \(x = 0\). From this point, you can apply the slope to identify other points on the line.

Graphing Linear Equations

Graphing linear equations is all about translating the equation into a visual format which is easy to understand at a glance. For the equation \(y = 3 - x\), first identify the slope and y-intercept using the slope-intercept form \(y = mx + b\). Once you have:

• The y-intercept (point) to start your graph, here (0, 3)
• The slope \(m = -1\) to determine the direction and steepness of the line

Next, graphing involves these steps:1. **Plot the y-intercept:** Go to the y-axis and mark where the line will cross, at (0, 3).2. **Use the slope to find another point:** Slope \(m = -1\) tells us to move one unit down and one to the right to find another point, resulting in (1, 2).3. **Draw the line:** Connect these points with a straight line. Extend it beyond these two points to visualize the line’s direction.By following these steps, you create a clear, comprehensive graph of the linear equation. Remember, each step meticulously builds on the one before, ensuring accuracy and clarity in your graphical representation.