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Chapter 5: Problem 65

Which line has the greater (a) Slope? (b) \(y\) -intercept? $$ y+2=3(x-1), \quad y=6-50 x $$

Short Answer

Expert verified
Answer: The line with the equation \(y=3x-5\) has the greater slope, and the line with the equation \(y=-50x+6\) has the greater y-intercept.

Step by step solution

01

Rewrite the first equation in slope-intercept form

To rewrite \(y+2=3(x-1)\) in slope-intercept form, we need to perform the following steps: 1. Distribute the 3 to \(x-1\): \(y+2=3x-3\) 2. Subtract 2 from both sides: \(y=3x-5\) Now, the first equation is written in slope-intercept form as \(y=3x-5\).
02

Rewrite the second equation in slope-intercept form

To rewrite the second equation, \(y=6-50x\), in slope-intercept form, we need to rewrite it in the form \(y=mx+b\). Currently, it is written as \(y=-50x+6\), which is already in slope-intercept form.
03

Identify the slopes and \(y\)-intercepts

Now that both equations are in slope-intercept form, we can compare their slopes and \(y\)-intercepts: 1. First equation: \(y=3x-5\), where the slope \(m_1=3\) and the \(y\)-intercept \(b_1=-5\) 2. Second equation: \(y=-50x+6\), where the slope \(m_2=-50\) and the \(y\)-intercept \(b_2=6\)
04

Compare the slopes and \(y\)-intercepts

(a) Compare the slopes: \(m_1=3\) and \(m_2=-50\). Since 3 > -50, the line with the equation \(y=3x-5\) has the greater slope. (b) Compare the \(y\)-intercepts: \(b_1=-5\) and \(b_2=6\). Since 6 > -5, the line with the equation \(y=-50x+6\) has the greater \(y\)-intercept.

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Key Concepts

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

Slope-Intercept Form
Understanding the slope-intercept form is crucial when working with linear equations. In this form, any linear equation can be expressed as \[y = mx + b\], where:
  • \(m\) is the slope,
  • \(b\) is the y-intercept.
This form reveals important information about lines:
  • The slope \(m\) shows the steepness and direction of the line.
  • The y-intercept \(b\) indicates where the line crosses the y-axis.
When a problem presents an equation not in slope-intercept form, such as \(y+2=3(x-1)\), rearranging it into this form helps uncover those details. This step involves expanding the equation and isolating \(y\) to match \[y = mx + b\].Through such transformations, the characteristics of the line become easier to analyze.
Linear Equations
Linear equations describe straight lines on a coordinate plane. They are called "linear" because their graph is a line. The structure of a linear equation allows you to predict the output \(y\) for any given input \(x\). Characteristics of linear equations include:
  • A constant rate of change, which is the slope.
  • The absence of variables raised to powers other than one.
Solving linear equations often involves converting them into a familiar format, like the slope-intercept form, to make comparisons or predictions. Every time you come across new equations, practice reformatting them into \[y = mx + b\]. This clarity enhances your understanding and aids in solving problems related to intersections, parallelism, and perpendicularity.
Y-Intercept
The y-intercept is the point where the graph of a line crosses the y-axis of a coordinate plane. In the context of the slope-intercept form \[y = mx + b\], it is represented by the constant \(b\). Key aspects of the y-intercept include:
  • It signifies the starting value of \(y\) when \(x = 0\).
  • Understanding the y-intercept helps in quickly sketching graphs of lines and determining their position relative to each other.
  • When comparing linear equations, the y-intercept indicates which line is higher or lower at the y-axis.
For instance, in the equations given in the original exercise, \(y=-50x+6\) has a y-intercept of 6, which clearly illustrates that this line crosses the y-axis higher than the line \(y=3x-5\), having a y-intercept of -5. Recognizing these differences is vital for identifying shifts or trends described by linear equations.

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Most popular questions from this chapter

Are the lines parallel? $$ y=7+4(x-2) ; y=8+2(2 x+3) $$

Which line has the greater (a) Slope? (b) \(y\) -intercept? $$ y=3+6 x, \quad y=5-3 x $$

Put the equation \(y=3 x t+2 x t^{2}+5\) in the form \(y=b+m x .\) What are the values of \(b\) and \(m ?\) [Note: Your answers could include \(t\).]

Write the equation in the form \(y=\) \(b+m x,\) and identify the values of \(b\) and \(m\). $$ A x+B y=C, \text { if } B \neq 0 $$

Decide for what value(s) of the constant \(A\) (if any) the equation has (a) The solution \(x=0\) (b) A positive solution in \(x\) (c) No solution in \(x\). $$ 3 x=A $$
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