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Chapter 1: Problem 85

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?

Short Answer

Expert verified
The x-intercept is \\( \left( -\frac{b}{m}, 0 \right) \\) when \\( m \neq 0 \\).

Step by step solution

01

Understanding the x-intercept

The x-intercept \((a, 0)\) is the point where a line crosses the x-axis. This means that the y-coordinate is zero. To find the x-intercept of the line equation \(y = mx + b\), we set \(y = 0\) and solve for \(x\).
02

Setting Up the Equation

Set \(y = 0\) in the equation \(y = mx + b\). This results in the equation \(0 = mx + b\). We need to arrange this equation to solve for \(x\).
03

Solving for x

Start with the equation \(0 = mx + b\). 1. Subtract \(b\) from both sides to get: \( -b = mx\). 2. Divide both sides by \(m\) to isolate \(x\): \( x = -\frac{b}{m}\).
04

Condition for a Single x-intercept

The equation \(x = -\frac{b}{m}\) is valid if \(m eq 0\) because division by zero is undefined. Therefore, for a single x-intercept to exist, the slope \(m\) must not be zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations are fundamental in algebra and are involved whenever you see a straight line on a graph. These are equations of the first degree, usually characterized by the standard form: \( y = mx + b \). Here:
  • \( y \) is the dependent variable representing the vertical position on a graph.
  • \( x \) is the independent variable representing the horizontal position.
  • \( m \) is the slope, showing the steepness of the line.
  • \( b \) is the y-intercept, indicating where the line crosses the y-axis.
Linear equations are everywhere, from calculating costs in business to predicting trends. Understanding them helps you model and solve real-life problems. When graphed, these equations form a straight line, hence the name 'linear.' This straight line indicates a constant rate of change between \( x \) and \( y \).
x-intercepts
An x-intercept of a line is a critical concept in algebra, as it defines where the line crosses the x-axis on a graph. For the line equation \( y = mx + b \), you find the x-intercept by setting \( y = 0 \).Here are the steps to find the x-intercept:
  • Start with the equation \( y = mx + b \).
  • Set \( y \) to 0: Thus, the equation becomes \( 0 = mx + b \).
  • Rearrange to solve for \( x \): Subtract \( b \) to get \( -b = mx \), then divide by \( m \): \( x = -\frac{b}{m} \).
However, remember that the slope \( m \) must not be zero. If \( m = 0 \), the line is horizontal and does not cross the x-axis, meaning that there would be no x-intercept. Thus, a line has a unique x-intercept as long as it is not horizontal (\( m eq 0 \)).
Slope of a Line
The slope of a line \( m \) in a linear equation is a measure of its inclination. It essentially tells us how steep the line is. A slope is a ratio that compares the change in the vertical direction to the change in the horizontal direction between any two points on a line.Here鈥檚 how to interpret slope:
  • If \( m > 0 \), the line is rising, moving upward from left to right.
  • If \( m < 0 \), the line is falling, moving downward from left to right.
  • If \( m = 0 \), the line is perfectly horizontal, indicating no change as \( x \) increases.
In the context of x-intercepts, it is important to note that \( m \) should not be zero for a line to intersect the x-axis except in degenerate cases where it is the only point of intersection. Slope not only affects the direction of a line, but it also determines whether the line will ever cross the x-axis again.

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