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Chapter 3: Problem 66

Describe what the graph of each linear equation will look like in the coordinate plane. (Hint: Rewrite the equation if necessary so that it is in a more recognizable form.) $$ x+4=3 $$

Short Answer

Expert verified
The graph is a vertical line at \( x = -1 \).

Step by step solution

01

Simplify the Equation

First, simplify the given equation. Start by subtracting 4 from both sides of the equation: \[ x + 4 - 4 = 3 - 4 \] This simplifies to: \[ x = -1 \]
02

Understand the Equation Form

The equation \( x = -1 \) is in a very specific form. This form represents a vertical line because it sets the value of \( x \) as \(-1\) for any value of \( y \).
03

Plot the Equation

To plot this equation on the coordinate plane, draw a vertical line that passes through the point \( x = -1 \). This line will extend infinitely in the positive and negative \( y \)-directions.

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Key Concepts

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

coordinate plane
The coordinate plane is a two-dimensional plane used to locate points. It consists of two perpendicular lines: the horizontal axis called the x-axis and the vertical axis called the y-axis. These axes intersect at a point called the origin, which has coordinates (0, 0). The plane is divided into four quadrants:
  • Quadrant I: both x and y are positive.
  • Quadrant II: x is negative and y is positive.
  • Quadrant III: both x and y are negative.
  • Quadrant IV: x is positive and y is negative.
Coordinates are written as (x, y), where 'x' denotes the horizontal distance from the origin and 'y' denotes the vertical distance.
vertical line
In the coordinate plane, a vertical line runs straight up and down. It has an undefined slope, meaning it doesn't rise or fall in the y-direction as it moves horizontally. A vertical line is represented by an equation of the form x = a where 'a' is a constant. For instance, the equation x = -1 means that the line is vertical and crosses the x-axis at -1. No matter what the value of y is, x always remains -1. To graph a vertical line, simply identify the x-coordinate where the line will cross the x-axis and draw a line parallel to the y-axis through that point.
plotting equations
Plotting equations involves graphing equations on the coordinate plane to visualize their behavior. There are steps to follow:

  • Step 1: Simplify the equation, if necessary, to recognize its form.
  • Step 2: Identify the type of line or curve. For linear equations, it could be a horizontal, vertical, or slanted line.
  • Step 3: Determine key points that the graph will pass through, such as intercepts.
  • Step 4: Draw the line or curve based on these points.
For example, in the exercise with the equation x + 4 = 3, simplifying this to x = -1 gives us a vertical line. To plot it:

  • Identify that x is constant at -1.
  • Draw a straight vertical line through x = -1 on the x-axis.
  • Extend the line infinitely in both positive and negative y-directions.
This is a complete graph of the equation x = -1.

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Most popular questions from this chapter

\(5 x-3 y=-2\) \(3 x-5 y=-8\)

Describe what the graph of each linear equation will look like in the coordinate plane. (Hint: Rewrite the equation if necessary so that it is in a more recognizable form.) $$ 3 x=9 y $$

Demand for an item is often closely related to its price. As price increases, demand decreases, and as price decreases, demand increases. Suppose demand for a video game is 2000 units when the price is 40 dollar, and demand is 2500 units when the price is 30 dollar. (a) Let \(x\) be the price and \(y\) be the demand for the game. Graph the two given pairs of prices and demands on the given grid. (b) Assume that the relationship is linear. Draw a line through the two points from part (a). From the graph, estimate the demand if the price drops to \(\$ 20\). (c) Use the graph to estimate the price if the demand is 3500 units. (d) Write the prices and demands from parts (b) and (c) as ordered pairs.

Graph each linear equation. $$ y=-1 $$

The height \(y\) (in centimeters) of a woman can be approximated by the linear equation $$ y=3.9 x+73.5 $$ where \(x\) is the length of her radius bone in centimeters. (a) Use the equation to approximate the heights of women with radius bones of lengths \(20 \mathrm{~cm}, 22 \mathrm{~cm},\) and \(26 \mathrm{~cm}\). (b) Write the information from part (a) as three ordered pairs. (c) Graph the equation for \(x \geq 20\), using the data from part (b). (d) Use the graph to estimate the length of the radius bone in a woman who is \(167 \mathrm{~cm}\) tall. Then use the equation to find the length of the radius bone to the nearest centimeter.
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