Question

The equation of a struight line parallel te the x-axis is given by (o) \(\frac{x-a}{1}=\frac{y-b}{1}=\frac{x-c}{1}\) (b) \(\frac{x-a}{0}=\frac{y-b}{1}=\frac{x-c}{1}\) (c) \(\frac{x-a}{0}=\frac{y-b}{0}=\frac{r-c}{1}\) (d) \(\frac{x-4}{1}=\frac{y-b}{0}=\frac{z-c}{0}\).

Short answer

Option (d) represents the equation of a line parallel to the x-axis.

Step-by-step solution

Identify the Condition for Parallel to X-axis

A line parallel to the x-axis has a constant y-value, which means the slope of the line is zero. Therefore, it takes the form: \( y = k \), where \( k \) is a constant.

Examine Each Option for Parallelism

For each option, substitute differentials to determine if the y-component remains constant, ensuring the line is parallel to the x-axis.

Analyze Option (a)

Option (a) is \( \frac{x-a}{1} = \frac{y-b}{1} = \frac{z-c}{1} \). This equation shows that it's not maintaining a constant y-value since both x and y are in the same ratio function, suggesting the line isn't parallel to the x-axis.

Analyze Option (b)

Option (b) is \( \frac{x-a}{0} = \frac{y-b}{1} = \frac{z-c}{1} \). Here, \( \frac{x-a}{0} \) is undefined, therefore this option does not represent a line at all and cannot be parallel to any axis.

Analyze Option (c)

Option (c) is \( \frac{x-a}{0} = \frac{y-b}{0} = \frac{z-c}{1} \). Both \( \frac{x-a}{0} \) and \( \frac{y-b}{0} \) are undefined, meaning this doesn't describe a line parallel or otherwise.

Analyze Option (d)

Option (d) is \( \frac{x-4}{1} = \frac{y-b}{0} = \frac{z-c}{0} \). The term \( \frac{y-b}{0} \) suggests that \( y \) is independent of both x and z, indicating that \( y = b \) is a constant, hence the line is parallel to the x-axis.

Key concepts

Parallel to X-Axis

When we talk about a line being parallel to the x-axis, it means that the line runs horizontally. Imagine the x-axis as a giant straight road. Any line parallel to it will follow the same straight, flat path without ever crossing or snagging away from it. The defining feature of such a line is its unchanging y-value.

You can identify these lines easily in the equation form:

• The equation is written as \( y = k \), where \( k \) is a constant y-value.
• The x-value changes, but the y-value remains constant.
• The slope of these lines is zero since there's no vertical change.

So, if you see a line where the y-value doesn't change, you've spotted a line parallel to the x-axis.

Undefined Slope

In mathematics, the slope of a line measures how steep the line is. When the slope is called 'undefined', it is a special case usually involving a vertical line. In the examined exercise, terms like \( \frac{x-a}{0} \) or \( \frac{y-b}{0} \) manifest situations leading to undefined slopes.

Here's what undefined slope implies:

• A vertical line where all x-values are the same but y-values may vary.
• The concept of 'rise over run' (change in y over change in x) falls apart when the run is zero, as you cannot divide by zero.
• Represented as an equation in the form \( x = a \), where \( a \) is a constant.

While undefined slopes pop up often in various problems, remember they typically indicate a vertical line, not parallel to the x-axis.

Constant Y-Value

A constant y-value is one of the simplest components in linear equations. This means a line remains at the same vertical level across any value of x. It's as if you drew a straight line across a page at a constant height.

Characteristics include:

• The y-value in the equation does not change as x changes.
• Indicates the line will never ascend or descend but remains horizontal.
• Equation presented as \( y = k \), where \( k \) is the constant y-value.

In our solution, finding that the y-term holds constant confirms the line's parallelism to the x-axis.

Line Parallelism

The concept of parallelism in lines is rooted in their feature of never crossing each other. Parallel lines have either the same slope or, in the case of horizontal lines, a slope of zero.

Important points about parallel lines:

• Two lines with identical slopes are parallel. For horizontal lines, this means each has a slope of zero.
• They maintain a constant distance from each other no matter how far they are extended.
• If they share the same slope and different y-intercepts, they are parallel on a plane.

Understanding these properties helps when solving problems that require identifying or working with parallel lines, like those discussed in the exercise.