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Chapter 3: Problem 10

Find the slope and the \(y\) -intercept of the line with the given equation. $$y=7$$

Short Answer

Expert verified
The slope of the line described by the equation \(y = 7\) is 0 and the \(y\)-intercept is 7.

Step by step solution

01

Understand the form of the equation

The given equation, \(y = 7\), is known as a horizontal line equation. Since there is no 'x' term in the equation, we can see that the slope of the line is 0.
02

Find the y-intercept

The \(y\)-intercept is the value of \(y\) when \(x = 0\). In our equation \(y = 7\), \(y\) stays at 7 regardless of the value of \(x\). Therefore, the \(y\)-intercept is 7.

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Key Concepts

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

Y-intercept
The term **y-intercept** refers to the point where a line crosses the y-axis on a graph. This point occurs when the value of \(x = 0\). For the equation of a line like \(y = 7\), determining the y-intercept is straightforward. Since every point on the line has a y-coordinate of 7, the y-intercept for this equation is simply \((0, 7)\).

  • The y-intercept is always expressed as a point: \((x, y)\).
  • It tells you where the line will hit the y-axis.
  • In horizontal line equations, the y-intercept is particularly clear, as the line maintains a constant y-value.
Horizontal Line Equation
A **horizontal line equation** is one where the y-value remains constant, and it does not depend on x. This means that the line doesn't tilt up or down, but runs straight across from left to right.

For the equation \(y = 7\), this indicates that no matter what value x takes on, y will always be 7. Such lines have a few distinct characteristics:
  • The slope of a horizontal line is 0, indicating no vertical change as you move across the x-axis.
  • Horizontal lines have no x-intercept, as they never cross the x-axis unless y equals 0.
  • They are represented simply by \(y = c\), where \(c\) is a constant value.
Horizontal lines are useful for depicting relationships where one variable remains unchanged regardless of another variable's values.
Linear Equations
**Linear equations** are mathematical expressions that describe a straight line on a graph. These equations are typically formatted as \(y = mx + b\), where \(m\) represents the slope (how steep the line is) and \(b\) represents the y-intercept.

However, the equation \(y = 7\) is slightly different鈥攊t is a special type of linear equation where \(m\) is 0 because there is no x term. Thus, it results in a horizontal line.

Some key features of linear equations include:
  • They graph as straight lines.
  • They can represent direct relationships between two variables.
  • Their solutions can be easily visualized on a coordinate plane.
Remember, linear equations can vary from horizontal, vertical, or sloped lines, each demonstrating different types of relationships between variables.

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Most popular questions from this chapter

$$\text { Simplify: } \quad 3\left(12 \div 2^{2}-3\right)^{2}$$.

write each sentence as a linear equation in two variables. Then graph the equation. The \(y\) -variable is 3 more than the \(x\) -variable.

Use a graphing utility to graph in a standard viewing rectangle, \([-10,10,1]\) by \([-10,10,1]\). Then use the \([\text { TRACE }]\) feature to trace along the line and find the coordinates of two points. $$y=-3 x+2$$

determine whether each statement 鈥渕akes sense鈥� or 鈥渄oes not make sense鈥� and explain your reasoning. The graph that I'm looking at is U-shaped, so its equation cannot be of the form \(y=m x+b\)

Will help you prepare for the material covered in the first section of the next chapter. Is \((4,-1)\) a solution of both \(x+2 y=2\) and \(x-2 y=6 ?\)
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