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Chapter 11: Problem 297

In the following exercises, graph by plotting points. $$ y=4 x-3 $$

Short Answer

Expert verified
Plot (-1, -7), (0, -3), (1, 1), and (2, 5) and connect them with a straight line.

Step by step solution

01

- Identify the Equation

The given equation is a linear equation in the form of y = mx + c, where m is the slope and c is the y-intercept.
02

- Determine Key Points

Choose several values for x to find corresponding y-values. Typically, using values such as -1, 0, 1, and 2 are useful.
03

- Calculate Coordinates

Substitute the chosen x-values into the equation to find the y-values.1. If x = -1: y = 4(-1) - 3 = -4 - 3 = -72. If x = 0: y = 4(0) - 3 = -33. If x = 1: y = 4(1) - 3 = 4 - 3 = 14. If x = 2: y = 4(2) - 3 = 8 - 3 = 5
04

- Plot the Points

Plot the calculated points (-1, -7), (0, -3), (1, 1), and (2, 5) on the Cartesian plane.
05

- Draw the Line

Connect the points with a straight line. This line represents the graph of the equation y = 4x - 3.

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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 is a fundamental concept in algebra. These equations are in the form of \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. The goal is to represent the equation visually on a graph. By doing so, we can better understand the relationship between the variables.
In the provided exercise, the equation \(y = 4x - 3\) is given. To graph this equation, we follow these steps:
  • Identify the form of the equation (linear).
  • Select several x-values to find corresponding y-values. Ideally, use values like -1, 0, 1, and 2.
  • Calculate the y-values by substituting the x-values into the equation.
  • Plot the resulting points on a graph.
  • Draw a line through these points to visualize the equation.
By plotting multiple points, it becomes clear how the line behaves, and how the variables interact.
Cartesian Plane
The Cartesian plane is a two-dimensional plane used to graph equations. It consists of two perpendicular axes:
  • The horizontal axis, called the x-axis.
  • The vertical axis, called the y-axis.
Together, these axes divide the plane into four quadrants. Each point on the plane is identified by an ordered pair \((x, y)\). In this exercise:
  • Points such as (-1, -7), (0, -3), (1, 1), and (2, 5) are plotted.
  • These points are identified by their coordinates on the x and y axes.
  • By marking these points and connecting them, we create the graph of the equation.
Understanding the Cartesian plane is essential for anyone learning to graph equations. It provides a clear and useful framework for visualizing mathematical relationships.
Coordinate System
A coordinate system is a method used to locate points in space. In a two-dimensional coordinate system, also known as the Cartesian coordinate system:
  • Each point is represented by an ordered pair \((x, y)\), where x is the horizontal position and y is the vertical position.
  • This system allows us to plot points easily and trace out graphs of equations.
For the exercise with the equation \(y = 4x - 3\):
  • We can select x-values and calculate their corresponding y-values.
  • These calculated pairs, like (-1, -7) and (2, 5), represent points that are plotted on the Cartesian plane.
By connecting these plotted points, we visualize the linear equation as a straight line. Understanding the coordinate system is crucial for graphing and provides an intuitive way to explore mathematical concepts.

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Most popular questions from this chapter

In the following exercises, plot each point in a rectangular coordinate system. $$(2,5),(5,2)$$

In the following exercises, graph the vertical and horizontal lines. $$ x=3 $$

In the following exercises, plot each point in a rectangular coordinate system. $$(1,3),(3,1)$$

In the following exercises, graph by plotting points. $$ \frac{1}{3} x+y=2 $$

In the following exercises, graph the vertical and horizontal lines. $$ x=-5 $$
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