Problem 8
Question
In Exercises 5 through 14, find an equation of the line satisfying the given conditions. $$ \text { Through the point }(1,-7) \text { and parallel to the } x \text { axis. } $$
Step-by-Step Solution
Verified Answer
The equation of the line is \(y = -7\).
1Step 1: Identify the characteristics of the line
The problem states that the line is parallel to the x-axis and passes through the point \(1, -7\). A line parallel to the x-axis is a horizontal line.
2Step 2: Determine the general form of the equation
A horizontal line has the form y = c, where c is a constant.
3Step 3: Substitute the given point into the equation
Since the line passes through the point \(1, -7\), substitute \(-7\) for \(y\) in the equation y = c. Thus, the equation is y = -7.
Key Concepts
Horizontal LinesParallel LinesEquation of a Line
Horizontal Lines
A horizontal line is a line that runs parallel to the x-axis. This means that no matter how far you go along the x-axis, the y-coordinate remains the same. You can visualize it as a straight line that goes from left to right.
The general equation of a horizontal line is simple: it is expressed as \(y = c\), where \(c\) is a constant value representing the y-coordinate of every point on the line. If you see an equation like \(y = -7\), it tells you that this is a horizontal line that crosses the y-axis at -7.
To summarize:
The general equation of a horizontal line is simple: it is expressed as \(y = c\), where \(c\) is a constant value representing the y-coordinate of every point on the line. If you see an equation like \(y = -7\), it tells you that this is a horizontal line that crosses the y-axis at -7.
To summarize:
- Horizontal lines are parallel to the x-axis
- They have the same y-coordinate for all points
- The equation is \(y = c\)
Parallel Lines
Parallel lines are two or more lines in a plane that never intersect. They have the same slope but different y-intercepts. This means they are always the same distance apart.
For example, consider the lines \(y = 3x + 2\) and \(y = 3x - 4\). Since both lines have the slope of 3, they are parallel.
When lines are parallel to the x-axis, they are also horizontal lines. Thus, their equations will always be in the form \(y = c\), where each line's constant is different but the slope (which in the case of horizontal lines is zero) is the same.
Key points:
For example, consider the lines \(y = 3x + 2\) and \(y = 3x - 4\). Since both lines have the slope of 3, they are parallel.
When lines are parallel to the x-axis, they are also horizontal lines. Thus, their equations will always be in the form \(y = c\), where each line's constant is different but the slope (which in the case of horizontal lines is zero) is the same.
Key points:
- Parallel lines never intersect
- They have identical slopes
- Horizontal parallel lines have equations like \(y = c_1\) and \(y = c_2\), where \(c_1\) and \(c_2\) are different constants
Equation of a Line
The equation of a line describes all the points that lie on that line. There are several forms to express this equation.
Slope-Intercept Form: The most common is the slope-intercept form, \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. For horizontal lines, since the slope (\(m\)) is 0, this reduces to \(y = b\).
Point-Slope Form: Another form is the point-slope form: \(y - y_1 = m(x - x_1)\), where \(m\) is the slope, and \( (x_1, y_1) \) is a specific point on the line. Again, for horizontal lines, this simplifies to \(y = y_1\).
Standard Form: You might also encounter the standard form \(Ax + By = C\). For horizontal lines, \(B\) would be non-zero and \(A\) would be zero, converting it to \(By = C\), or simply \(y = C/B\).
Key formats:
Slope-Intercept Form: The most common is the slope-intercept form, \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. For horizontal lines, since the slope (\(m\)) is 0, this reduces to \(y = b\).
Point-Slope Form: Another form is the point-slope form: \(y - y_1 = m(x - x_1)\), where \(m\) is the slope, and \( (x_1, y_1) \) is a specific point on the line. Again, for horizontal lines, this simplifies to \(y = y_1\).
Standard Form: You might also encounter the standard form \(Ax + By = C\). For horizontal lines, \(B\) would be non-zero and \(A\) would be zero, converting it to \(By = C\), or simply \(y = C/B\).
Key formats:
- Slope-Intercept Form: \(y = mx + b\)
- Point-Slope Form: \(y - y_1 = m(x - x_1)\)
- Standard Form: \(Ax + By = C\)
Other exercises in this chapter
Problem 8
In Exercises 7 through 28 , draw a sketch of the graph of the equation. $$ y=4 x-3 $$
View solution Problem 8
Prove that the points \(A(6,-13), B(-2,2), C(13,10)\), and \(D(21,-5)\) are the vertices of a square. Find the length of a diagonal.
View solution Problem 9
In Exercises 7 through 12, the functions \(f\) and \(g\) are defined. In each problem define the following functions and determine the domain of the resulting f
View solution Problem 9
In Exercises 1 through 10 , find the domain and range of the given function, and draw a sketch of the graph of the function. $$ \phi=\\{(x, y)|y=| 3 x+2 \mid\\}
View solution