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Chapter 3: Problem 27

Graph each linear equation. Plot four points for each line. $$y=3$$

Short Answer

Expert verified
Draw a horizontal line passing through y = 3, for any values of x.

Step by step solution

01

Understand the Equation

The given equation is a horizontal line because it is in the form of y = constant. In this case, the equation is y = 3.
02

Choose Values for x

For a horizontal line, the y-value remains constant, so we need to choose different values for x to plot the points. Let's choose x = -2, x = 0, x = 2, and x = 4.
03

Find Corresponding y-values

Since the line is horizontal at y = 3, the value of y is always 3 for any value of x. Thus, at x = -2, y = 3; at x = 0, y = 3; at x = 2, y = 3; and at x = 4, y = 3.
04

Plot the Points on the Graph

Plot the points (-2, 3), (0, 3), (2, 3), and (4, 3) on the coordinate plane.
05

Draw the Line

Draw a straight horizontal line through all the plotted points, which represents the equation y = 3.

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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 mathematical expressions that form straight lines when graphed on a coordinate plane. They can be simplified to the form \[y = mx + b\], where \(m\) is the slope and \(b\) is the y-intercept. Understanding linear equations helps us predict how values are related.Key points to note:
  • They produce straight lines.
  • The slope \(m\) tells how steep the line is.
  • The intercept \(b\) shows where the line crosses the y-axis.

Linear equations are fundamental in algebra and are used to model real-world problems.
coordinate plane
The coordinate plane, also called the Cartesian plane, is a two-dimensional surface where we plot points, lines, and curves. It consists of two axes:

The horizontal axis is called the x-axis, and the vertical axis is the y-axis. These axes intersect at a point called the origin (0,0).

We use the coordinate plane to show relationships visually. Each point on the plane is represented by a pair of numbers \((x, y)\), called coordinates. The first number specifies the position along the x-axis, and the second number specifies the position along the y-axis.
  • Understand the scale of the axes before plotting.
  • Make sure to label your axes.

The coordinate plane is a powerful tool in mathematics to visualize equations and other mathematical concepts.
horizontal lines
Horizontal lines are unique because they have a slope \(m\) of 0, meaning they stay flat and do not slant up or down. They are in the form of \[ y = c\], where \(c\) is a constant. For example, the equation \( y = 3 \) represents a horizontal line passing through all points where the y-coordinate is 3.

Characteristics of horizontal lines:
  • They run parallel to the x-axis.
  • Their slope is always zero.
  • Every point on the line has the same y-value.

Horizontal lines are easy to identify and graph. Simply plot points where the y-coordinate is the same and draw a straight line through these points.
For instance, for the line \( y = 3 \), you would plot points like \((-2, 3)\), \((0, 3)\), \((2, 3)\) and \((4, 3)\).
plotting points
Plotting points is the first step in graphing linear equations. A point on the coordinate plane is defined by an ordered pair \((x, y)\), where \(x\) represents the horizontal position and \(y\) represents the vertical position.

To plot points correctly:
  • Start from the origin \((0,0)\).
  • Move horizontally along the x-axis to the x-coordinate.
  • Then, move vertically to the y-coordinate.
  • Mark the point where these two positions intersect.

For instance, to plot the point \((2, 3)\):
Start at the origin, move 2 units right along the x-axis, and then 3 units up along the y-axis. Place a dot at this position. Repeat the same steps for other points.
Effective plotting ensures accuracy in graphing, making it crucial for visualizing linear equations and their solutions accurately.

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Most popular questions from this chapter

Determine whether each pair of lines is parallel, perpendicular, or neither. $$2 y=x+6, y-2 x=4$$

Solve each problem. See Example 9. Depth and flow. When the depth of the water in the Tangipahoa River at Robert, Louisiana, is 9.14 feet, the flow is 1230 cubic feet per second ( \(\mathrm{ft}^{3} / \mathrm{sec}\) ). When the depth is 7.84 feet, the flow is \(826 \mathrm{ft}^{3} / \mathrm{sec} .\) (U.S. Geological Survey, www.usgs.gov). Let \(w\) represent the flow in cubic feet per second and \(d\) represent the depth in feet. a) Write the equation of the line through \((9.14,1230)\) and \((7.84,826)\) and express \(w\) in terms of \(d .\) Round to two decimal places. b) What is the flow when the depth is \(8.25 \mathrm{ft} ?\) c) Is the flow increasing or decreasing as the depth increases?

Determine whether each pair of lines is parallel, perpendicular, or neither. $$2 x-4 y=9, \frac{1}{3} x=\frac{2}{3} y-8$$

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) has \(x\) -intercept \((2,0)\) and \(y\) -intercept \((0,4)\).

Solve each problem. See Example 9. Carbon dioxide emission. Worldwide emission of carbon dioxide (CO \(_{2}\) ) increased linearly from 14 billion tons in 1970 to 26 billion tons in 2000 (World 91影视 Institute, www.wri.org). a) Express the emission as a linear function of the year in the form \(y=m x+b,\) where \(y\) is in billions of tons and \(x\) is the year. [ Hint: Write the equation of the line through \((1970,14) \text { and }(2000,26) .]\) b) Use the function from part (a) to predict the worldwide emission of \(\mathrm{CO}_{2}\) in 2010 .
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