Problem 27
Question
In Exercises 17-28, find the slope and \(y\)-intercept (if possible) of the equation of the line. Sketch the line. \( x + 5 = 0 \)
Step-by-Step Solution
Verified Answer
The given equation \(x + 5 = 0\) does not have a standard slope or y-intercept, because it's a vertical line. This line lies at \(x = -5\) and does not cross the y-axis.
1Step 1: Find the Slope
For a general line equation \(y = mx + b\), \(m\) is the slope. In the given equation \(x + 5 = 0\), there is no \(y\) term, and the equation shows no change in \(y\) for any change in \(x\), therefore the slope is undefined.
2Step 2: Find the y-intercept
The y-intercept is the point at which the line crosses the y-axis. However, since the equation \(x + 5 = 0\) lacks a \(y\) term, there's no traditional y-intercept. This line is vertical and does not cross the y-axis.
3Step 3: Plot the Line
Rewriting the equation \(x + 5 = 0\) as \(x = -5\), tells us that the line is a vertical one at \(x = -5\) on the x-axis. It runs parallel to y-axis and hence, does not intersect the y-axis.
Key Concepts
Slope of a LineY-InterceptVertical Line EquationUndefined SlopeGraphing Linear Equations
Slope of a Line
Understanding the slope of a line is crucial in algebra and geometry. The slope indicates how steep a line is and the direction it goes. For a linear equation in the form of
\( y = mx + b \)
, the coefficient
\( m \)
represents the slope. It's calculated by the rise over run formula, which is the change in the
\( y \)
values divided by the change in the
\( x \)
values between two distinct points on the line. A positive slope means the line is inclining upward, while a negative slope indicates a downward incline. A horizontal line has a slope of zero, meaning there is no rise over any run. Conversely, a vertical line, as in the exercise, has an undefined slope, because the change in
\( x \)
is zero, thus making the division by zero undefined.
\( y = mx + b \)
, the coefficient
\( m \)
represents the slope. It's calculated by the rise over run formula, which is the change in the
\( y \)
values divided by the change in the
\( x \)
values between two distinct points on the line. A positive slope means the line is inclining upward, while a negative slope indicates a downward incline. A horizontal line has a slope of zero, meaning there is no rise over any run. Conversely, a vertical line, as in the exercise, has an undefined slope, because the change in
\( x \)
is zero, thus making the division by zero undefined.
Y-Intercept
The
\( y \)
-intercept of a line is the point where the line crosses the
\( y \)
-axis, which corresponds to when
\( x = 0 \)
. In the equation
\( y = mx + b \)
, the
\( b \)
term represents this y-intercept. To find it, we set
\( x \)
to zero and solve for
\( y \)
. However, for a vertical line, finding a
\( y \)
-intercept is not applicable because the line never touches the
\( y \)
-axis, and thus does not have a specific point where
\( y \)
is isolated.
\( y \)
-intercept of a line is the point where the line crosses the
\( y \)
-axis, which corresponds to when
\( x = 0 \)
. In the equation
\( y = mx + b \)
, the
\( b \)
term represents this y-intercept. To find it, we set
\( x \)
to zero and solve for
\( y \)
. However, for a vertical line, finding a
\( y \)
-intercept is not applicable because the line never touches the
\( y \)
-axis, and thus does not have a specific point where
\( y \)
is isolated.
Vertical Line Equation
A vertical line has a specific form, which is
\( x = k \)
, where
\( k \)
is a constant and represents the position of the line parallel to the
\( y \)
-axis. Vertical lines are unique because they have an undefined slope and no
\( y \)
-intercept, and every point on the line has the same
\( x \)
-coordinate. In our original exercise, the line
\( x + 5 = 0 \)
simplifies to
\( x = -5 \)
, indicating a vertical line situated at
\( x = -5 \)
on the coordinate plane.
\( x = k \)
, where
\( k \)
is a constant and represents the position of the line parallel to the
\( y \)
-axis. Vertical lines are unique because they have an undefined slope and no
\( y \)
-intercept, and every point on the line has the same
\( x \)
-coordinate. In our original exercise, the line
\( x + 5 = 0 \)
simplifies to
\( x = -5 \)
, indicating a vertical line situated at
\( x = -5 \)
on the coordinate plane.
Undefined Slope
When discussing slopes, it's important to understand what it means for a slope to be undefined. This situation occurs exclusively with vertical lines, as the formula for slope
\( (\Delta y / \Delta x) \)
involves dividing by zero, which is mathematically undefined. The concept of undefined slope resonates with the fact that vertical lines do not have a slope that can be quantified in the same way other lines are—try to remember, vertical equals undefined slope!
\( (\Delta y / \Delta x) \)
involves dividing by zero, which is mathematically undefined. The concept of undefined slope resonates with the fact that vertical lines do not have a slope that can be quantified in the same way other lines are—try to remember, vertical equals undefined slope!
Graphing Linear Equations
Graphing linear equations involves plotting lines on the coordinate plane based on their equations. The general steps include finding the slope and y-intercept (if they exist) and then using these elements to draw the line. When given an equation of the form
\( y = mx + b \)
, you can easily plot the y-intercept by locating point
\( b \)
on the y-axis, and use the slope
\( m \)
to find another point. For non-conventional forms, like vertical or horizontal lines, identifying their distinctive features (for example, a vertical line's constant
\( x \)
value or a horizontal line's constant
\( y \)
value) is the primary step in plotting. All lines have a place on the graph, though how they are represented differs depending on their properties.
\( y = mx + b \)
, you can easily plot the y-intercept by locating point
\( b \)
on the y-axis, and use the slope
\( m \)
to find another point. For non-conventional forms, like vertical or horizontal lines, identifying their distinctive features (for example, a vertical line's constant
\( x \)
value or a horizontal line's constant
\( y \)
value) is the primary step in plotting. All lines have a place on the graph, though how they are represented differs depending on their properties.
Other exercises in this chapter
Problem 27
In Exercises 23-32, find the zeros of the function algebraically. \(f(x) = \frac{1}{2}x^3 - x\)
View solution Problem 27
In Exercises 19-36, determine whether the equation represents \(y\) as a function of \(x\). \(y^2 = x^2 - 1\)
View solution Problem 27
In Exercises 23-32, find the \( x \)- and \( y \)-intercepts of the graph of the equation. \( y = |3x-7| \)
View solution Problem 27
In Exercises 27-38, find the distance between the points. \( (6, -3) \), \( (6, 5) \)
View solution