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Chapter 3: Problem 19

Write an equation of the line with each given slope, \(m\), and \(y\) -intercept, \((0, b) .\) $$ m=0, b=-8 $$

Short Answer

Expert verified
The equation of the line is \(y = -8\).

Step by step solution

01

Identify the Equation Form

The equation of a line is generally given in the form of the slope-intercept equation: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
02

Substitute the Slope

Substitute the given slope \(m = 0\) into the equation \(y = mx + b\). This results in \(y = 0x + b\).
03

Substitute the Y-Intercept

Now, substitute the given y-intercept \(b = -8\) into the equation, replacing \(b\) in \(y = 0x + b\). This gives us \(y = 0x - 8\).
04

Simplify the Equation

Since \(0x\) is equal to zero, the equation simplifies to \(y = -8\). Thus, the equation of the line is \(y = -8\).

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Key Concepts

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

Slope-Intercept Form
A line can be represented using the slope-intercept form, which is very useful in algebra for writing the equation of a line. This form is expressed as \( y = mx + b \). In this equation:
  • \( m \) represents the slope of the line.
  • \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
The slope-intercept form makes it easy to graph a linear equation or write the equation of a line if you know the slope and y-intercept. For example, in the equation \( y = x + 4 \), the slope \( m \) is 1, and the y-intercept \( b \) is 4. Knowing the slope-intercept form helps in quick visualization and understanding of the line's behavior.
Slope of a Line
The slope of a line, commonly denoted as \( m \), measures the steepness or incline of the line. It shows how much the line rises or falls vertically for every unit it moves horizontally.
  • If the slope is positive, the line goes upwards as it moves from left to right.
  • If the slope is negative, the line goes downwards as it moves from left to right.
  • A zero slope indicates a perfectly horizontal line.
  • An undefined slope means a vertical line.
The formula to find the slope if two points on the line are known, \((x_1, y_1)\) and \((x_2, y_2)\), is \( m = \frac{y_2-y_1}{x_2-x_1} \). When the slope is zero, as in \( y = -8 \), the line is horizontal and does not rise or fall at all.
This results in a constant value for \( y \), indicating a flat line along the y-axis.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. This is a crucial feature of linear equations as it specifies where the line starts on the graph.
  • It is represented by \( b \) in the slope-intercept equation \( y = mx + b \).
  • The coordinates of the y-intercept are \((0, b)\).
Understanding the y-intercept helps in graphing the line quickly, as it gives a fixed starting point on the vertical axis. For instance, in the exercise \( y = -8 \), the y-intercept is \(-8\). This means the line crosses the y-axis at \( y = -8 \) and remains constant along a horizontal path.

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Most popular questions from this chapter

For Exercises 71 through 75, fill in each blank with "0," "positive," or "negative." For Exercises 76 and 77, fill in each blank with "x"or "y." Point \(\quad\) (_____,_____) Location \(\quad\) origin

Determine whether each pair of lines is parallel, perpendicular, or neither. See Example 7. $$ \begin{array}{l} y=\frac{1}{5} x+20 \\ y=-\frac{1}{5} x \end{array} $$

Mixed Practice Find the slope of each line. See Examples 3 through 6. $$ -4 x-7 y=9 $$

Write an equation of the line with each given slope, \(m\), and \(y\) -intercept, \((0, b) .\) $$ m=-4, b=-\frac{1}{6} $$

Write an ordered pair for each point described. Find the area of the rectangle whose vertices are the points with coordinates \((5,2),(5,-6),(0,-6),\) and (0,2) .
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