Since slope is calculated using the formula [tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex], and line [tex]z+b[/tex], the slope of both lines is equivalent to ________. It is given that the lines are parallel, and we calculated that the slopes are the same. Therefore, parallel lines have the same slopes.
[tex]\frac{z-v}{x-w}[/tex]
[tex]\frac{x-w}{z-v}[/tex]
[tex]\frac{v-z+b}{x-z+a}[/tex]
[tex]\frac{w-x+a}{v-z+b}[/tex]
Answer (2)
Calculate the slope of the line passing through points ( x , z ) and ( w , v ) using the slope formula: m = x − w z − v .
Recognize that parallel lines have the same slope.
Equate the slopes of both lines: m 1 = m 2 .
Conclude that the slope of both lines is x − w z − v .
Explanation
Problem Analysis The problem states that the slope is calculated using the formula m = x 2 − x 1 y 2 − y 1 . We have two lines: one passing through points ( x , z ) and ( w , v ) , and another line given as z + b . The lines are parallel, which means they have the same slope. We need to find the expression that represents the slope of both lines.
Calculate the slope of the first line First, let's calculate the slope of the line passing through the points ( x , z ) and ( w , v ) using the slope formula: m = x − w z − v This is the slope of the first line.
Analyze the second line The second line is given by the equation z + b . However, this equation is incomplete as it is missing a variable. To find the slope of the second line, we need two points on that line. Since the problem is not well-defined, we will assume that the question meant to give us the points ( x , z ) and ( w , v ) for the first line and ask for the slope of this line.
Determine the slope of the second line Since the lines are parallel, their slopes are equal. The slope of the first line is x − w z − v . Therefore, the slope of the second line is also x − w z − v .
Select the correct option Now, let's examine the provided options and determine which one matches the calculated slope:
The first option is x − w z − v , which matches the slope of the first line.
The second option is z − v x − w , which is the inverse of the slope.
The third option is x − z + a v − z + b .
The fourth option is v − z + b w − x + a .
Since the lines are parallel, their slopes are equal. The slope of the first line is x − w z − v . Therefore, the correct answer is x − w z − v .
Examples
In architecture, parallel lines are often used in building design, such as in the design of walls or beams. Knowing that parallel lines have the same slope is crucial for ensuring structural stability and aesthetic consistency. For example, if you're designing a roof with parallel beams, calculating the slope ensures that the beams are aligned correctly, distributing weight evenly and preventing structural failure.
The slope of a line through points ( x , z ) and ( w , v ) is calculated as x − w z − v . Since parallel lines have the same slope, the slope of both lines is therefore x − w z − v . Thus, the correct answer is the first option, x − w z − v .
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