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Chapter 0: Problem 31

Find an equation for each line. Then write your answer in the form \(A x+B y+C=0 .\) With \(y\) -intercept 3 and slope 2

Short Answer

Expert verified
The equation in standard form is \( 2x - y + 3 = 0 \).

Step by step solution

01

Understand the Slope-Intercept Form

First, let's recall the slope-intercept form of a line's equation, which is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Substitute the Given Slope and Y-Intercept

Given the slope \( m = 2 \) and the y-intercept \( b = 3 \), substitute these values into the slope-intercept form: \( y = 2x + 3 \).
03

Rearrange into Standard Form

To convert the equation \( y = 2x + 3 \) into the standard form \( Ax + By + C = 0 \), first subtract \( y \) from both sides to get \( 0 = 2x - y + 3 \).
04

Adjust the Signs for Standard Form

Multiply the entire equation by -1 to maintain the standard form structure of \( Ax + By + C = 0 \) with positive \( A \): \( 2x - y + 3 = 0 \).
05

Final Standard Form Equation

Confirm that the line's equation in standard form is \( 2x - y + 3 = 0 \).

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a common way of writing the equation of a line. It's written as \[ y = mx + b \]where:
  • \( m \) is the slope of the line, representing the rate of change or how steep the line is.
  • \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
This form is particularly useful because it gives you a clear visual representation of the line in terms of its slope and its starting point on the y-axis. For instance, if you're given that the slope \(m = 2\) and the y-intercept \(b = 3\), you can readily write the equation as:\[ y = 2x + 3 \]This equation means that for every unit increase in \(x\), \(y\) increases by 2, and the line crosses the y-axis at 3.
Standard Form of a Line
The standard form of a line's equation provides a different representation as compared to slope-intercept form. It is expressed as:\[ Ax + By + C = 0 \]where \(A\), \(B\), and \(C\) are integers and \(A\) should preferably be a positive value. Here's why standard form can be so useful:
  • It allows for easy calculation of intercepts and can be used to handle vertical and horizontal lines effortlessly.
  • It's ideal for use in algebraic operations, like finding intersections.
To convert from slope-intercept form to standard form, rearrange the equation \[ y = 2x + 3 \]by moving all terms to one side to get:\[ 2x - y + 3 = 0 \]Lastly, always check the signs to ensure \(A\) is positive as this is typically desired in standardized mathematics problems.
Y-intercept
The y-intercept is one of the defining characteristics of a line on a graph. It is the point where the line crosses the y-axis. When written as a coordinate, it appears as \((0, b)\), where \(b\) is the y-value.In the context of finding the equation of a line, the y-intercept is particularly important because it is an inherent part of the slope-intercept form \(y = mx + b\).For example, knowing that the y-intercept is 3 gives you the point \((0, 3)\),meaning the line touches the y-axis 3 units above the origin. Once you have the y-intercept, combined with a known slope, you can effectively describe the line's equation as \(y = 2x + 3\).This is an important step in moving from a visual representation to a precise mathematical equation.

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Most popular questions from this chapter

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(x^{2}+y^{2}=4\)

Let \(f(x)=\sqrt{16}-x\) and \(g(x)=x^{4}\). What is the domain of each of the following? (a) \(f\) (b) \(f \circ g\) (c) \(g \circ f\)

Sketch the graphs of \(f(x)=(x-3) / 2\) and \(g(x)=\sqrt{x}\) using the same coordinate axes. Then sketch \(f+g\) by adding \(y\) -coordinates.

For \(f(x)=1 /(x+1)-1 / x,\) find each value (if possible). (a) \(f(1)\) (b) \(f\left(-\frac{1}{2}\right)\) (c) \(f(-1)\) (d) \(f(t-1)\) (e) \(f\left(\frac{1}{t}\right)\)

Prove that the operation of composition of functions is associative; that is, \(f_{1} \circ\left(f_{2} \circ f_{3}\right)=\left(f_{1} \circ f_{2}\right) \circ f_{3}\).
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