Problem 87 - Brief 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 \((-\frac{b}{m}, 0)\) and exists if \(m \neq 0\).

Step 1: Understanding the Concept

An \(x\)-intercept is a point where the line crosses the \(x\)-axis, meaning at this point, the \(y\)-coordinate is 0. Thus, we find the \(x\)-intercept by setting \(y = 0\) in the equation of the line.

Step 2: Setting Up the Equation

Given the line equation \(y = mx + b\), substitute \(y = 0\) to find the \(x\)-coordinate. Thus, we have the equation: \[0 = mx + b\]

Step 3: Solving for \(x\)

Rearrange the equation \(0 = mx + b\) to express \(x\): \[mx = -b\]Divide both sides by \(m\) to solve for \(x\): \[x = -\frac{b}{m}\]

Step 4: Condition for Existence

For the line to have a single \(x\)-intercept, \(m\) (the slope) must not be zero. If \(m = 0\), the line is horizontal (\(y = b\)), and either it does not cross the \(x\)-axis (when \(b eq 0\)) or is the whole \(x\)-axis (when \(b = 0\)). Therefore, \(m eq 0\) ensures one \(x\)-intercept.

Key Concepts

x-intercept

The concept of an \(x\)-intercept is fundamental when studying linear equations. An \(x\)-intercept occurs where a graph crosses the \(x\)-axis. This point is defined by a coordinate in the form \((x, 0)\). The reason why zero appears as the \(y\)-coordinate is because at this specific point, the graph lies directly on the \(x\)-axis, where the \(y\)-value is always zero.

To find the \(x\)-intercept of a line with the equation \(y = mx + b\), you set \(y\) to zero and solve for \(x\). This is because the intercept occurs precisely where \(y\) equals zero. This simplification leads to the equation \(0 = mx + b\). Solving this equation helps find the precise \(x\)-coordinate of the intercept.

Understanding \(x\)-intercepts helps in graphing lines and understanding their position relative to the axes. It is an essential step in solving and manipulating linear equations.

slope-intercept form

The slope-intercept form of a linear equation is one of the most commonly used formulas because of its simplicity and efficiency in graphing linear equations. This form is written as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) stands for the y-intercept.

Here's why it's beneficial:

• Slope \(m\): The slope determines the steepness and direction of a line. If \(m\) is positive, the line slopes upwards. If \(m\) is negative, the line slopes downward.
• Y-Intercept \(b\): This is where the line crosses the \(y\)-axis. It's the point you can quickly plot on a graph as your starting point.

With these two pieces, you can draw any straight line easily.

The beauty of slope-intercept form is that it gives a quick snapshot of the line's behavior. You immediately know how steep the line is and where it starts on the \(y\)-axis.

solving linear equations

Solving linear equations involves finding the value of \(x\) that makes the equation true. For equations in slope-intercept form such as \(y = mx + b\), solving an equation might mean identifying where the line crosses the \(x\)-axis.

To solve for \(x\) in these scenarios, you will often rearrange and simplify the equation. For example, if you need to find the \(x\)-intercept, you set \(y = 0\) and rearrange the equation \(mx + b = 0\) to find \(x = -\frac{b}{m}\). This operation involves isolating \(x\) on one side of the equation.

Key steps when solving linear equations:

• Identify what you are solving for (e.g., x-intercept or when \(y\) equals a particular value).
• Perform operations to isolate the variable \(x\) (addition, subtraction, multiplication, division).
• Rearrange the equation correctly to find a precise \(x\)-value.

Solving these equations is a fundamental skill, and understanding the required operations helps you tackle various algebra problems efficiently.