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Chapter 2: Problem 33

Write an equation in slope-intercept form for the line that satisfies each set of conditions. \(x\) -intercept \(-4, y\) -intercept 4

Short Answer

Expert verified
The equation is \( y = x + 4 \).

Step by step solution

01

Understand the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. We need to find values for both \(m\) and \(b\) for the line in question.
02

Identify the y-intercept

The problem states that the y-intercept is 4. This means the equation can start as \(y = mx + 4\). Here, \(b = 4\).
03

Use the x-intercept to find the slope

The x-intercept is given as -4. At the x-intercept, the value of \(y\) is 0, so we can use it to find the slope by substituting into the equation. Substitute \(x = -4\) and \(y = 0\) into \(y = mx + 4\): \[ 0 = m(-4) + 4 \] This simplifies to: \[ -4m = -4 \] Solve for \(m\) to get \(m = 1\).
04

Write the Final Equation

Now that we have the slope \(m = 1\) and the y-intercept \(b = 4\), we can write the equation of the line in slope-intercept form: \[ y = 1x + 4 \] This simplifies further to \( y = x + 4 \).

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Key Concepts

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

Understanding x-intercept
The x-intercept of a line is the point where the line crosses the x-axis. This point has a characteristic feature: the y-value is always zero at the x-intercept. For a given line, knowing the x-intercept is extremely useful for writing equations of the line, especially in the slope-intercept form. In our example, the x-intercept is \(-4\), meaning the line crosses the x-axis at this point.
This tells us that when plotting the graph of the line, the line will intersect the x-axis at the point \((-4, 0)\).
  • This information can be directly used to determine other features of the line, such as the slope.
  • It helps in establishing that for x-intercept, \(y = 0\).
Understanding this intersection provides a concrete step towards determining the overall behavior and the equation of the line.
Exploring y-intercept
The y-intercept is equally crucial as the x-intercept when working with linear equations. The y-intercept of a line is where the line crosses the y-axis. At this point, the x-value is zero. In the context of slope-intercept form \(y = mx + b\), the y-intercept is denoted by \(b\).
  • For our particular problem, the y-intercept is given as \(4\), meaning the line crosses the y-axis at \((0, 4)\).
  • This value directly tells us the constant in our linear equation, indicating the starting point of the line when plotting a graph.
This knowledge helps set the baseline for analyzing the linear equation, which helps us understand the entire line's placement in the graph.
Demystifying Slope
The slope of a line is a measure of how steep or flat the line is. It's expressed as the ratio of the rise (change in y) over the run (change in x). In slope-intercept form, it is represented with \(m\). It's crucial because it depicts the direction and angle of the line.
In our example, after using the given x-intercept and y-intercept, we calculated the slope as \(1\). This means the line rises up one unit for every unit it moves to the right.
  • A slope of \(1\) indicates a line at a \(45^{\circ}\) angle in a standard coordinate system.
  • The slope also determines if a line is increasing or decreasing.
Grasping this concept assists immensely in plotting the line accurately and understanding its behavior across different values of x.
Linear Equations and Slope-Intercept Form
Linear equations are foundational in mathematics, describing lines in a two-dimensional space. The slope-intercept form is \(y = mx + b\), where \(m\) represents the slope and \(b\) stands for the y-intercept. This form is advantageous as it quickly tells us about the line's slope and where it crosses the y-axis.
In our task, the line's equation was simplified to \(y = x + 4\).
This means:
  • The slope \(m\) is \(1\), indicating a steady, 1 to 1 change between x and y.
  • The y-intercept \(b\) is \(4\), marking the starting height of the line on the y-axis.
Understanding these linear equations aids us in interpreting linear graphs and their correlation to real-world scenarios, enabling easier visualization and analysis.

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Most popular questions from this chapter

Government For Exercises \(50-53\) , use the table below that shows the number of members of the U.S. House of Representatives with 30 or more consecutive years of service in Congress from 1991 to \(2003 .\) $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {1991} & {1993} & {1995} & {1997} & {1999} & {2001} & {2003} \\ \hline 11 & {12} & {9} & {6} & {3} & {7} & {9} \\\ \hline\end{array} $$ Identify the domain and range. Determine whether the relation is discrete or continuous.

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