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Chapter 1: Problem 52

Show that the linear equation \(\frac{x}{a}+\frac{y}{b}=1\) has \(x\) -intercept \((a, 0)\) and \(y\) -intercept \((0, b)\). (The \(x\) -intercept is the point where the line crosses the \(x\) -axis.

Short Answer

Expert verified
X-intercept: \((a, 0)\); Y-intercept: \((0, b)\).

Step by step solution

01

Identify the X-Intercept

To find the x-intercept of the line, set the value of \(y\) to zero in the equation \(\frac{x}{a} + \frac{y}{b} = 1\). This is because, at the x-intercept, the line crosses the x-axis, where the y-coordinate is zero.
02

Calculate X-Intercept

Substitute \(y = 0\) into the equation: \(\frac{x}{a} + \frac{0}{b} = 1\), simplifying to \(\frac{x}{a} = 1\). Then, solve for \(x\): multiply both sides by \(a\) to get \(x = a\). Thus, the x-intercept is \((a, 0)\).
03

Identify the Y-Intercept

To find the y-intercept of the line, set \(x\) to zero in the equation \(\frac{x}{a} + \frac{y}{b} = 1\). This is because, at the y-intercept, the line crosses the y-axis, where the x-coordinate is zero.
04

Calculate Y-Intercept

Substitute \(x = 0\) into the equation: \(\frac{0}{a} + \frac{y}{b} = 1\), simplifying to \(\frac{y}{b} = 1\). Then, solve for \(y\): multiply both sides by \(b\) to get \(y = b\). Thus, the y-intercept is \((0, b)\).
05

Conclusion

The x-intercept of the given line is \((a, 0)\) and the y-intercept is \((0, b)\), as shown by substituting \(y = 0\) and \(x = 0\) into the original equation respectively.

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Key Concepts

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

Understanding the X-Intercept
The x-intercept of a linear equation is a fundamental concept. It describes the point where the graph of the equation crosses the x-axis. To find this important point, you set the y-value in the equation to zero. Why zero? Because on the x-axis, any point's y-coordinate is always zero. This remains true regardless of other components or transformations.
  • Set the equation: Start with the original equation \(\frac{x}{a} + \frac{y}{b} = 1\).
  • Substitute \(y = 0\): This simplifies our work because we only focus on x.
  • Reason: \(\frac{x}{a} + \frac{0}{b} = 1\) reduces to \(\frac{x}{a} = 1\).
Now, you solve for x. Simply multiply both sides by \(a\) and you get \(x = a\). This means the x-intercept is located at \((a, 0)\). Such simplicity helps in many practical applications where understanding intercepts is crucial. Each x-intercept tells us a specific point on the x-axis that the line reaches.
Understanding the Y-Intercept
Similar to the x-intercept, the y-intercept holds key insights into the behavior of linear equations. It's where the line crosses the y-axis, meaning the x-coordinate is zero. It's useful in graphing and understanding real-world contexts like starting points in various scenarios.
  • Set the equation: Using \(\frac{x}{a} + \frac{y}{b} = 1\) again.
  • Substitute \(x = 0\): Now, focus on y.
  • Reason: \(\frac{0}{a} + \frac{y}{b} = 1\) simplifies to \(\frac{y}{b} = 1\).
To solve for y, multiply both sides by \(b\). You find \(y = b\), showing the y-intercept at \((0, b)\). In real-world applications, this tells you where, along the vertical axis, the line initiates when moving within a plane. This parameter is highly valued for deducing exact crossing points.
Understanding the Slope-Intercept Form
The slope-intercept form is a concise way to express linear equations. It is given by the formula \(y = mx + b\), where \(m\) is the slope, representing the rate of change, and \(b\) is the y-intercept. This form makes it easy to identify key features of the line at a glance.
  • Equation Format: Rearrange terms to better apply the slope-intercept form.
  • Identify Components: From \(\frac{x}{a} + \frac{y}{b} = 1\) to all variables.
  • Observation: Rearranging gives \(y = -\frac{b}{a}x + b\).
Thus, the slope \(m\) becomes \(-\frac{b}{a}\) and the y-intercept remains \(b\). This format allows easy plotting of the line, simplifying calculations. Understanding both the slope, which indicates steepness, and intercepts, becomes intuitive. This approach is highly favored in algebra for its straightforward application.

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Most popular questions from this chapter

For each equation, find the slope \(m\) and \(y\) intercept \((0, b)\) (when they exist) and draw the graph. \(y=2 x\)

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