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Chapter 7: Problem 20

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ y=-3 $$

Short Answer

Expert verified
y-intercept at (0, -3); no x-intercept. Graph is a horizontal line through y = -3.

Step by step solution

01

- Identify the Equation Type

The given equation is in the form of a horizontal line: \(y = -3\). This means the value of \(y\) is constant and always \(-3\).
02

- Find the y-Intercept

The y-intercept occurs where the line crosses the y-axis (where \(x = 0\)). For the equation \(y = -3\), when \(x = 0\), \(y\text{ is already }-3\). Therefore, the y-intercept is \((0, -3)\).
03

- Find the x-Intercept

The x-intercept occurs where the line crosses the x-axis (where \(y = 0\)). However, since the equation \(y = -3\) shows \(y\text{ is never }0\), there is no x-intercept.
04

- Graph the Equation

To graph \(y = -3\), draw a horizontal line that passes through the y-axis at \(y = -3\). This line extends infinitely to the left and right.

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Key Concepts

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

X-Intercept
The x-intercept of a graph is the point where the line crosses the x-axis. To find the x-intercept, we set the value of y to 0 and solve for x. However, not all equations will have an x-intercept. In cases where the graph is a horizontal line parallel to the x-axis, like in our example where the equation is \(y = -3\), the line never touches or crosses the x-axis. This is because the value of y is constant and does not change to 0, hence no x-intercept. Understanding the absence of an x-intercept in such cases is crucial for correctly interpreting graphs and equations involving horizontal lines.
Y-Intercept
The y-intercept is the point at which the line crosses the y-axis. This happens when x is set to 0. To find the y-intercept, simply substitute x with 0 in the given equation. For the equation \(y = -3\), when \(x = 0\) the corresponding \(y\) value is \-3\. Thus, the y-intercept is \( (0, -3) \). This means the line intersects the y-axis at this point. Identifying the y-intercept helps in plotting the initial point of the line on the graph, making it an essential step in graphing linear equations.
Horizontal Line
A horizontal line is a straight line that extends left and right across the graph. It has a constant y value and does not slant or tilt. The general form of the equation of a horizontal line is \(y = c\), where \ c \ is a constant. In our example, \(y = -3\), the value of y is always -3, regardless of the x value. This means that for every value of x, from negative to positive infinity, the line will remain at \ y = -3 \ on the graph. To graph a horizontal line, find the y-intercept and draw a straight line that runs horizontally through this point. Horizontal lines are straightforward to graph and analyze, as their simplicity underscores fundamental concepts in linear equations.

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Most popular questions from this chapter

Determine whether each pair of lines is parallel, perpendicular, or neither $$ 2 x+5 y=-7 \text { and } 5 x-2 y=1 $$

Determine whether each pair of lines is parallel, perpendicular, or neither $$ 2 x=y+3 \text { and } 2 y+x=3 $$

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Solve each system by the elimination method. Check each solution. $$ \begin{array}{l} \frac{1}{3} x+\frac{1}{2} y=\frac{13}{6} \\ \frac{1}{2} x-\frac{1}{4} y=-\frac{3}{4} \end{array} $$
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