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Chapter 3: Problem 59

Graph each equation. $$3 y=9$$

Short Answer

Expert verified
The graph of the equation \(y=3\) is a straight horizontal line passing through the point (0,3).

Step by step solution

01

Simplify the Equation

Divide both sides of the equation by 3, resulting in the simplified form of the equation: \(y=3\).
02

Identify Key Characteristics of the Graph

The graph is a horizontal line that cuts the y-axis at 3. This is because y is always 3, no matter what value x takes.
03

Graph the Equation

Draw a horizontal line through the point (0,3) on the y-axis. This line 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.

Horizontal Lines
When we speak about horizontal lines in graphing linear equations, these lines are both intriguing and simple to understand. A horizontal line on a graph means that no matter the value of \( x \), the value of \( y \) remains constant. In other words, you are dealing with an equation where \( y \) appears alone on one side, such as \( y = 3 \).

The beauty of horizontal lines is in their consistency:
  • They always run parallel to the x-axis.
  • The slope of a horizontal line is 0, which signifies no vertical change as you move along the line.
  • This stability makes them easy to graph!
For \( y = 3 \), the line will intersect the y-axis at the point (0,3) and extend indefinitely in both directions along the x-axis.
Simplifying Equations
Simplifying equations is a crucial step in the process of graphing linear equations. It's vital to reduce any complex equation into its simplest form to make the graphing process straightforward. Let's consider our example equation, \( 3y = 9 \).

To simplify, simply isolate \( y \) on one side of the equation. This typically involves operations such as:
  • Dividing every term by a common factor
  • Combining like terms, if needed
In this instance, divide both sides by 3 to achieve the simplified form \( y = 3 \).

Simplified equations not only make graphing easier, but they also assist in identifying key characteristics of a graph, such as slopes, intercepts, and overall behavior.
y-intercept
The y-intercept is a fundamental aspect of understanding and graphing linear equations. This is the specific point where the line crosses the y-axis. For horizontal lines like \( y = 3 \), determining the y-intercept is straightforward because the equation tells us directly where the line cuts the y-axis.

A few pointers about y-intercepts:
  • For any equation of the form \( y = b \), the y-intercept is \( b \).
  • It is represented on the graph by the point \( (0, b) \).
  • This single point is vital for setting up your graph accurately.

In our previous example, since the equation simplifies to \( y = 3 \), the line crosses the y-axis at (0, 3), establishing this point as the y-intercept. Understanding y-intercepts helps to quickly ascertain a line鈥檚 starting position on the y-axis, making graphing much clearer.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. In each exercise, evaluate $$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$ for the given ordered pairs \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\). $$\left(x_{1}, y_{1}\right)=(3,4) ;\left(x_{2}, y_{2}\right)=(5,4)$$

Use a graphing utility to graph each equation.Then use the \([\text { TRACE }]\) feature to trace along the line and find the coordinates of two points. Use these points to compute the line's slope. $$y=\frac{3}{4} x-2$$

Write an equation in slope-intercept form of the line satisfying the given conditions. The line has an \(x\) -intercept at \(-4\) and is parallel to the line containing \((3,1)\) and \((2,6)\)

Exercises \(82-84\) will help you prepare for the material covered in the next section. In each exercise, solve for \(y\) and put the equation in slope- intercept form. $$y+3=-\frac{3}{2}(x-4)$$

Solve for \(h: \quad V=\frac{1}{3} A h .\) (Section 2.4, Example 4)
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