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Chapter 3: Problem 58

Concept Check In Exercises \(55-62,\) 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

- Rewrite the equation

First, rewrite the given equation in a simpler form. Given the equation is: \( x + 4 = 3 \) Subtract 4 from both sides to solve for \( x \): \( x + 4 - 4 = 3 - 4 \) This simplifies to: \( x = -1 \).
02

- Understand the Simplified Equation

The equation \( x = -1 \) represents a vertical line in the coordinate plane. In this form, \( x \) is always the same value, regardless of the value of \( y \).
03

- Draw the Graph

To visualize the graph, plot a vertical line through \( x = -1 \) on the coordinate plane. This line will extend infinitely in the up and down direction but will not move left or right from \( x = -1 \).

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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 surface where we can plot points, lines, and curves to represent equations visually. It is made up of two perpendicular lines:
  • The x-axis (horizontal line)
  • The y-axis (vertical line)
The point where these two axes intersect is called the origin, labeled \( (0,0) \). Each point on the plane is defined by a pair of numbers known as coordinates. The first number represents the position on the x-axis, while the second number represents the position on the y-axis, written as \( (x,y) \).
For example, in the simplified form of our original equation \( x = -1 \), no matter the value of y, the x-coordinate is always -1. This concept helps us to graph the line accurately on the coordinate plane.
Vertical Line
A vertical line is a straight line that moves up and down but does not move left or right. In a coordinate plane, a vertical line has an equation of the form \( x = k \), where \( k \) is a constant. In our example, we simplified the equation to \( x = -1 \). This tells us that the line crosses the x-axis at -1 and that every point on this line has an x-coordinate of -1.
To graph this line:
  • Locate -1 on the x-axis
  • Draw a line parallel to the y-axis passing through this point
This visualization helps us understand that regardless of the y-coordinate, the x-coordinate remains -1.
Simplifying Equations
Simplifying an equation means changing it into a more basic or recognizable form without altering its meaning. This is a crucial step in understanding and solving many math problems.
In the original equation \( x + 4 = 3 \), we simplified it by isolating x on one side. Here鈥檚 the process in detail:
  • Start with the original equation: \( x + 4 = 3 \)
  • Subtract 4 from both sides to isolate x: \( x + 4 - 4 = 3 - 4 \)
  • This simplifies to: \( x = -1 \)
By breaking down the equation in this way, it becomes easier to identify the type of line or graph represented in the coordinate plane.
Solving for x/y
When solving linear equations, it is common to solve for one variable in terms of the other. This means isolating either x or y on one side of the equation to understand how they relate to each other.
In our example, we solved for x:
  • The original equation: \( x + 4 = 3 \)
  • Simplify by subtracting 4 from both sides: \( x = -1 \)
When an equation is in this form, it鈥檚 easy to graph. The equation \( x = -1 \) shows that x is always -1, and y can be any value.
Similarly, if an equation required solving for y, the process would involve isolating y and simplifying the equation appropriately.
Linear Equations
Linear equations form the backbone of algebra and graphing. They represent straight lines on a coordinate plane. A general form of a linear equation is \( Ax + By = C \).
In our example, the equation simplified to \( x = -1 \), which is a special type of linear equation representing a vertical line.
Key characteristics of linear equations:
  • They graph as straight lines
  • Their highest exponent of the variable is always 1
  • They can be written in various forms: slope-intercept form (\( y = mx + b \)), standard form (\( Ax + By = C \)), or simplified forms (\( x = k \) or \( y = k \))
Understanding linear equations is essential as they are used in various applications, from simple graph plotting to complex problem-solving.

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Most popular questions from this chapter

Solve each problem. The table lists the average annual cost (in dollars) of tuition and fees at 2 -year colleges for selected years, where year 1 represents \(2004,\) year 2 represents \(2005,\) and so on. \(\begin{array}{|c|c|}\hline \text { Year } & {\text { cost }(\text { in dollars })} \\ {1} & {2079} \\ {2} & {2182} \\ {3} & {2272} \\ {4} & {2361} \\ {5} & {2402} \\ \hline\end{array}\) (a) Write five ordered pairs from the data. (b) Plot the ordered pairs. Do the points lie approximately in a straight line? (c) Use the ordered pairs \((1,2079)\) and \((4,2361)\) to write an equation of a line that approximates the data. Give the final equation in slope-intercept form. (d) Use the equation from part (c) to estimate the average annual cost at 2 -year colleges in \(2009 \text { to the nearest dollar. (Hint: What is the value of } x \text { for } 2009 ?)\)

Graph each equation by using the slope and y-intercept. $$ y=3 x+2 $$

Write an equation of the line satisfying the given conditions. Give the final answer in slope intercept form. (Hint: Recall the relationships among slopes of parallel and perpendicular lines in Section \(3.3 .)\) Through \((4,2) ;\) perpendicular to \(x-3 y=7\)

Evaluate each expression. $$ 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 $$

Concept Check In Exercises \(55-62,\) 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.) $$ 2 x=y-4 $$
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