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Chapter 1: Problem 16

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. \(y=2 x\)

Short Answer

Expert verified
Slope \( m = 2 \), \( y \)-intercept is (0, 0). Graph is a line through these points.

Step by step solution

01

Identifying the Equation Form

The given equation is in the form of a linear equation, specifically the slope-intercept form: \( y = mx + b \). In this form, \( m \) represents the slope and \( b \) represents the \( y \)-intercept.
02

Finding the Slope \( m \)

In the equation \( y = 2x \), the coefficient of \( x \) is the slope \( m \). Thus, \( m = 2 \). The slope tells us that for every 1 unit increase in \( x \), \( y \) increases by 2 units.
03

Finding the \( y \)-intercept \( b \)

In the equation \( y = 2x \), since there is no constant term added or subtracted, the \( y \)-intercept \( b \) is 0. This means the graph crosses the \( y \)-axis at the origin, (0, 0).
04

Graphing the Equation

Start by plotting the \( y \)-intercept (0, 0) on the graph. Since the slope \( m = 2 \), which is \( \frac{2}{1} \), from the point (0, 0), move up 2 units and right 1 unit to locate another point on the graph (1, 2). Draw a straight line through these points to complete the graph.

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Key Concepts

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

graphing linear equations
Graphing linear equations can seem complex at first, but it becomes straightforward once you understand the process. A linear equation describes a straight line on the coordinate plane. This is because the change between any two points on the line is uniform or constant. To graph a linear equation, we typically utilize information such as the slope and the y-intercept.
Simply follow these steps to graph a linear equation:
  • Identify the slope and y-intercept from the equation. These will give you the starting point and direction in which to draw the line.
  • Start by plotting the y-intercept on the graph. This is the point where the line crosses the y-axis.
  • From the y-intercept, use the slope to find another point on the graph. The slope tells you how many units to move up or down for every unit you move left or right.
  • Draw a line connecting these points. This line is the graphical representation of your equation.
By following these steps, graphing linear equations becomes a systematic and manageable task.
slope-intercept form
The slope-intercept form is one of the most common ways to represent linear equations. It is known for its simplicity and ease of use in graphing. The form is written as:

\( y = mx + b \)

Here, \( m \) represents the slope of the line, and \( b \) represents the y-intercept.
  • The **slope** (\( m \)) indicates how steep the line is. A larger absolute value of \( m \) means a steeper line. If \( m \) is positive, the line rises as you move from left to right; if negative, it falls.
  • The **y-intercept** (\( b \)) is the point where the line crosses the y-axis. This is where \( x = 0 \).
The beauty of the slope-intercept form is that it directly provides both the slope and the y-intercept, making the task of graphing much simpler.
finding slope and y-intercept
Finding the slope and y-intercept from a linear equation is a crucial skill, especially when dealing with equations in the slope-intercept form.
Let's take the equation \( y = 2x \) as an example.
  • **Finding the Slope**: In the equation \( y = mx + b \), the coefficient of \( x \) is the slope \( m \), which is 2 in this case. This means for every unit increase in \( x \), \( y \) will increase by 2 units.
  • **Finding the Y-intercept**: The constant term in the equation represents the y-intercept \( b \). Since there is no constant term apart from \( 2x \), the y-intercept is \( b = 0 \). This indicates that the line will pass through the origin (0,0).
Recognizing these components allows you to quickly understand and graph the equation on a coordinate plane.

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Most popular questions from this chapter

Use your graphing calculator to graph the following four equations simultaneously on the window \([-10,10]\) by \([-10,10]:\) $$ \begin{array}{l} y_{1}=2 x+6 \\ y_{2}=2 x+2 \\ y_{3}=2 x-2 \\ y_{4}=2 x-6 \end{array} $$ a. What do the lines have in common and how do they differ? b. Write the equation of another line with the same slope that lies 2 units below the lowest line. Then check your answer by graphing it with the others.

GENERAL: Seat Belt Use Because of driver education programs and stricter laws, seat belt use has increased steadily over recent decades. The following table gives the percentage of automobile occupants using seat belts in selected years. $$ \begin{array}{lcccc} \hline \text { Year } & 1995 & 2000 & 2005 & 2010 \\ \hline \text { Seat Belt Use (\%) } & 60 & 71 & 81 & 86 \\ \hline \end{array} $$ a. Number the data columns with \(x\) -values \(1-4\) and use linear regression to fit a line to the data. State the regression formula. [Hint: See Example 8.] b. Interpret the slope of the line. From your answer, what is the yearly increase? c. Use the regression line to predict seat belt use in \(2015 .\) d. Would it make sense to use the regression line to predict seat belt use in 2025 ? What percentage would you get?

A 5 -foot-long board is leaning against a wall so that it meets the wall at a point 4 feet above the floor. What is the slope of the board? [Hint: Draw a picture.]

Give two definitions of slope.

ATHLETICS: Muscle Contraction The fundamental equation of muscle contraction is of the form \((w+a)(v+b)=c\), where \(w\) is the weight placed on the muscle, \(v\) is the velocity of contraction of the muscle, and \(a, b\), and \(c\) are constants that depend upon the muscle and the units of measurement. Solve this equation for \(v\) as a function of \(w, a, b\), and \(c\).
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