Problem 91 - Applied Calculus

Question

Find the \(x\) -intercept \((a, 0)\) where the line \(y=m x+b\) crosses the \(x\) -axis. Under what condition on \(m\) will a single \(x\) -intercept exist?

Step-by-Step Solution

Answer

The x-intercept is \(x = \frac{-b}{m}\). A single x-intercept exists if \(m \neq 0\).

Step 1: Understanding x-intercept

The x-intercept is the point where the line crosses the x-axis. At this point, the value of y is zero since it is on the x-axis.

Step 2: Set up the equation

Since the x-intercept occurs where y is zero, substitute y = 0 into the equation y = mx + b. This gives us 0 = mx + b.

Step 3: Solve for x

Re-arrange the equation 0 = mx + b to solve for x. Subtract b from both sides to get mx = -b. Then divide both sides by m to solve for x, resulting in x = -b/m.

Step 4: Condition for a single x-intercept

For there to be a single x-intercept, the slope m must not be zero. If m = 0, the line is horizontal and does not intersect the x-axis, unless b = 0, which would make the entire line coincide with the x-axis. Thus, a single x-intercept exists when m ≠ 0.

Key Concepts

Line Equation

A line equation is a mathematical expression that represents a straight line in a coordinate plane. It describes the relationship between the x-values and y-values of points along the line. To find any point on the line, plug the x-coordinate into the equation, and solve for y. This will give you the corresponding y-coordinate. Similarly, you can find the x-coordinate by plugging the y-value into the equation and solving for x. Line equations are fundamental in algebra and geometry. They are widely used to solve problems related to slopes, intercepts, and the positions of lines on the graph.

Slope-Intercept Form

The slope-intercept form of a line equation is a simple way to express the equation of a line using the formula:\[ y = mx + b \]Here, \(m\) represents the slope of the line, and \(b\) indicates the y-intercept. The slope \(m\) tells us how steep the line is, and in which direction it slants. Meanwhile, the y-intercept \(b\) is the point where the line crosses the y-axis. The slope-intercept form is very user-friendly. It provides quick insight into the line's behavior and makes graphing easier:

• If \(m > 0\), the line slants upwards.
• If \(m < 0\), the line slants downwards.
• If \(b = 0\), the line passes through the origin.

Slope of a Line

The slope of a line measures the line's steepness. This crucial concept in coordinate geometry shows how much the y-value changes for a change in the x-value.The formula for calculating the slope \(m\) of a line between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula shows us the rise (change in y) over run (change in x). A larger slope indicates a steeper line.

• Horizontal lines have a slope of 0, which means they have no steepness.
• Vertical lines have an undefined slope because their run is 0.

Understanding slope helps determine how a line on a graph changes. It’s a cornerstone in the analysis of linear equations and systems.