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Chapter 1: Problem 48

Given an equation having \(x\) and \(y\) as variables, explain how to determine the \(x\) - and \(y\) -intercepts.

Short Answer

Expert verified
For \( ax + by = c \), x-intercept is \( (\frac{c}{a}, 0) \) and y-intercept is \( (0, \frac{c}{b}) \).

Step by step solution

01

Understand Intercepts

The x-intercept is the point where a graph crosses the x-axis. At this point, the y-coordinate is 0. Similarly, the y-intercept is where a graph crosses the y-axis, and the x-coordinate is 0.
02

Given Equation

Suppose we have a linear equation: \( ax + by = c \). To find the intercepts, we will evaluate this equation by setting x or y to 0.
03

Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( ax + by = c \). This simplifies to \( ax = c \). Solve for x by dividing both sides by \( a \), resulting in \( x = \frac{c}{a} \). Thus, the x-intercept is \( (\frac{c}{a}, 0) \).
04

Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( ax + by = c \). This simplifies to \( by = c \). Solve for y by dividing both sides by \( b \), resulting in \( y = \frac{c}{b} \). Hence, the y-intercept is \( (0, \frac{c}{b}) \).
05

Verify Results

Review to ensure that by substituting y=0, and x=0 we obtain the appropriate x-intercept and y-intercept on the given line equation.

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Key Concepts

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

Linear Equations
Linear equations form the backbone of numerous mathematical concepts and have significant applications in different fields such as physics, economics, and engineering. At its core, a linear equation is an algebraic expression that represents a straight line when graphed on a coordinate plane. These equations typically take the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.

Linear equations possess a few key characteristics:
  • They are 1st-degree equations, which means the highest power of the variable is 1.
  • The graph of a linear equation is always a straight line.
  • They can have one, none, or infinitely many solutions depending on the system of equations they are part of.
Understanding linear equations is essential because they help establish relationships between different variables. By grasping this foundation, interpreting graphs and real-world phenomena becomes a manageable task.
Graphing Intercepts
Graphing intercepts is a method used to help visualize linear equations by identifying points where the line crosses the axes. These points, known as intercepts, are essential in understanding the position and orientation of a line on a graph.

In simple terms, a graph has two key intercepts:
  • The **x-intercept** is where the line meets the x-axis. At this point, the y-value is always zero.
  • The **y-intercept** is where the line meets the y-axis. Here, the x-value is always zero.
To graph intercepts for a linear equation, such as \( ax + by = c \):
  • Find the x-intercept by setting \( y = 0 \) and solving for \( x \).
  • Discover the y-intercept by setting \( x = 0 \) and solving for \( y \).
These intercept points help draw the overall line by providing two specific locations where the line crosses the axes. This method simplifies graphing, as you only need two points to define a line.
Solving Equations
Solving equations is a fundamental skill in mathematics, especially when working with linear equations like \( ax + by = c \). The main aim is to find the values of variables (usually \( x \) and \( y \)) that make the equation true. Understanding the process will lead to a more profound comprehension of the behavior of graphs and intersecting lines.

Here's a step-by-step approach to solving linear equations:
  • First, identify the equation and isolate one variable by setting constraints. For intercepts, set \( x = 0 \) to find the y-intercept and \( y = 0 \) for the x-intercept.
  • Simplify the resulting equations to solve for the respective variable. For example, setting \( y = 0 \) in \( ax + by = c \) simplifies to \( ax = c \). Solving for \( x \) reveals the x-intercept as \( (\frac{c}{a}, 0) \).
  • Repeating the procedure but this time setting \( x = 0 \) helps find the y-intercept, which would be \( (0, \frac{c}{b}) \).
This approach not only solves equations but also helps derive key understanding about lines and their graphical representations. By practicing equation-solving, learners build problem-solving confidence and expertise, which is applicable across many areas of study.

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Most popular questions from this chapter

The table lists the worldwide average household spending (in dollars) on Apple products for selected years. $$\begin{array}{|l|c|c|c|c|} \hline \text { Year } & 2009 & 2011 & 2013 & 2015 \\\ \hline \begin{array}{c} \text { Spending } \\ \text { (\$ dollars) } \end{array} & 62 & 158 & 265 & 444 \end{array}$$ (a) Use regression to find a formula \(f(x)=a x+b\) so that \(f\) models the data. (b) Interpret the slope of the graph of \(y=f(x)\) (c) Estimate the average household spending on Apple products in 2014 and compare it with the actual value of \(\$ 343\)

The table lists the average tuition and fees (in constant 2010 dollars) at public colleges and universities for selected years. $$\begin{array}{|l|l|l|l|l|l|}\hline \text { Year } & 1980 & 1990 & 2000 & 2005 & 2010 \\\\\hline \begin{array}{l}\text { Tuition and Fees } \\\\\text { (in 2010 dollars) }\end{array} & 5938 & 7699 & 9390 & 11,386 & 13,297 \\\\\hline\end{array}$$ (a) Find the equation of the least-squares regression line that models the data. (b) Graph the data and the regression line in the same viewing window. (c) Estimate tuition and fees in 2007 . (d) Use the model to predict tuition and fees in 2016 .

Worldwide gambling revenue from online betting was \(\$ 18\) billion in 2007 and \(\$ 24\) billion in \(2010 .\) (Source: Christiansen Capital Advisors.) (a) Find an equation of a line \(y=m x+b\) that models this information, where \(y\) is in billions of dollars and \(x\) is the year. (b) Use this equation to estimate online betting revenue in 2013.

Using the variable \(x\), write each interval using set-builder notation. $$[2,7)$$

Sketch the graph of \(f\) by hand. $$f(x)=-\frac{2}{3} x$$
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