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Chapter 2: Problem 87

Determine whether each pair of lines is parallel, perpendicular, or neither. \(x=6\) and \(6-x=8\)

Short Answer

Expert verified
The lines are parallel.

Step by step solution

01

Identify the Form of the First Line

The first equation is given as a vertical line, represented by the equation \(x = 6\). A vertical line has an undefined slope.
02

Rewrite the Second Equation

The second equation is given as \(6 - x = 8\). First, solve for \(x\): \[ 6 - x = 8 \] Subtract 6 from both sides: \[ -x = 2 \] Multiply both sides by -1 to obtain: \[ x = -2 \] This is also a vertical line.
03

Determine the Relationship Between the Lines

Both lines have equations of the form \(x = \text{constant}\), making them vertical lines. Vertical lines are parallel to each other.

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Key Concepts

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

Vertical Lines
Vertical lines are one of the two most basic forms of lines in coordinate geometry. A vertical line runs straight up and down and can be represented by an equation of the form \( x = c \) where \( c \) is a constant.
For example, the line \( x = 6 \) is vertical and passes through all points where the x-coordinate is 6. The y-coordinates can be any value, but the x-coordinate is always fixed at 6.
Identifying these types of lines is crucial because vertical lines always have an undefined slope, which brings us to our next topic:
Undefined Slope
The slope of a line is a measure of its steepness and is calculated as the ratio of the rise (change in y) over the run (change in x). However, in the case of vertical lines like \( x = 6 \), the 'run' is zero because the x-coordinate doesn't change, meaning you'd divide by zero鈥攕omething mathematically undefined.
So, vertical lines have what we call an undefined slope. When you encounter an equation in the form of \( x = c \), you can immediately determine that it represents a vertical line with an undefined slope.
Understanding slope helps in determining the relationship between lines, especially when verifying if they are parallel or perpendicular.
Equation Solving
Solving an equation involves finding the value of the variable that makes the equation true. In the step by step solution above, we saw the equation \(6 - x = 8 \). To find \( x \), follow these steps:
  • First, isolate the x-term by subtracting 6 from both sides: \[ 6 - x - 6 = 8 - 6 \] Simplifies to: \[ -x = 2 \]
  • Next, solve for \( x \) by multiplying both sides by -1: \[ -x \times (-1) = 2 \times (-1) \] Simplifies to: \[ x = -2 \]

Therefore, the equation \( 6 - x = 8 \) simplifies to \( x = -2 \). Just like \( x = 6 \), the equation \( x = -2 \) represents a vertical line with an undefined slope. Hence, both are vertical lines and hence parallel.

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Most popular questions from this chapter

Graph each line passing through the given point and having the given slope. (0,-4)\(; m=-\frac{3}{2}\)

Find the midpoint of each segment with the given endpoints. $$ (-9,3) \text { and }(9,8) $$

Three points that lie on the same straight line are said to be collinear. Consider the points \(A(3,1), B(6,2),\) and \(C(9,3) .\) Use the slope formula to determine whether the points \((1,-2),(3,-1),\) and (5,0) are collinear.

Graph the intersection of each pair of inequalities. $$ 2 x-y \geq 2 \text { and } y<4 $$

Determine whether each relation defines \(y\) as a function of \(x\). \(y=\sqrt{2 x+9}\)
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