/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 33 Find the slope of each line. \... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 33

Find the slope of each line. \(y=5 x-2\)

Short Answer

Expert verified
The slope is 5.

Step by step solution

01

Identify the Line Equation Form

The given equation is in the slope-intercept form, which is written as \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) represents the y-intercept.
02

Extract the Slope

In the given equation \( y = 5x - 2 \), identify the coefficient of \( x \), which is the slope \( m \). Here \( m = 5 \).
03

Confirm the Solution

Ensure that the equation is correctly interpreted. The slope \( m \) corresponds to the value you identified, which is \( 5 \).

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Key Concepts

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

Linear Equations
Linear equations are expressions that create straight lines when graphed on a coordinate plane. These equations are fundamental in algebra and represent the relationship between two variables, typically noted as x and y.
In the slope-intercept form, a linear equation is expressed as \( y = mx + b \). This form makes it easy to read off the slope and the y-intercept directly from the equation.
  • Straight Line: A linear equation always results in a straight line graph.
  • Two Variables: Typically involves an x and y variable showing their dependency.
  • Simple Format: Slope-intercept form \( y = mx + b \) is widely used for its simplicity and clarity.
Recognizing the format of linear equations can help in quickly identifying the slope and y-intercept, which are crucial for graphing.
Slope
The concept of slope is critical in understanding how a line behaves on a graph. Simply put, the slope of a line measures its steepness or incline.
In the context of a linear equation in slope-intercept form \( y = mx + b \), the slope is represented by \( m \). This value tells you how much y changes for a one-unit increase in x.
  • Positive Slope: The line rises as it moves from left to right, as seen in our example, since \( m = 5 \).
  • Negative Slope: A line falls as you move along from left to right when the slope is negative.
  • Zero Slope: Results in a horizontal line, highlighting no change in y as x changes.
  • Infinite Slope: Reflected by a vertical line onde the slope is undefined.
Understanding the slope helps predict how changes in the x-value affect the y-value, which is crucial for solving and interpreting linear equations.
Y-Intercept
The y-intercept is a vital component of the linear equation that provides specific information about the line's positioning. It indicates the point where the line crosses the y-axis.
In the slope-intercept form \( y = mx + b \), the y-intercept is represented by \( b \). This value gives the point \( (0, b) \) on the graph, meaning that when x equals zero, y is equal to \( b \).
  • Intersection Point: The y-intercept is the line's intersection with the y-axis.
  • Direct Value: In our example \( y = 5x - 2 \), the y-intercept is \( -2 \), showing the line crosses the y-axis at \( (0, -2) \).
  • Fixed Point: Unlike the slope, the y-intercept gives a fixed position rather than a measure of steepness.
Identifying the y-intercept is essential when you need to graph a linear equation or understand where it lies in connection to the axes.

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Most popular questions from this chapter

Find the slope of the line through the given points. \((2.1,6.7)\) and \((-8.3,9.3)\)

Write an ordered pair for each point described. Three vertices of a rectangle are \((-2,-3),(-7,-3),\) and \((-7,6)\) a. Find the coordinates of the fourth vertex of a rectangle. b. Find the perimeter of the rectangle. c. Find the area of the rectangle.

Solve. See the Concept Check in this section. Is the graph of \((3,0)\) in the same location as the graph of \((0,3) ?\) Explain why or why not.

Given two points on a nonvertical line, explain how to use the point-slope form to find the equation of the line.

Answer each exercise with true or false. The ordered pair \(\left(2, \frac{2}{3}\right)\) is a solution of \(2 x-3 y=6\)
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