What is the equation of the line that passes through $(1,2)$ and is parallel to the line whose equation is $2x+y-1=0$?
A. $2x+y+4=0$
B. $2x+y-4=0$
C. $2x-y-4=0
Answer (1)
Find the slope of the given line 2 x + y − 1 = 0 , which is − 2 .
Since the desired line is parallel, it has the same slope, − 2 .
Use the point-slope form y − y 1 = m ( x − x 1 ) with the point ( 1 , 2 ) and slope − 2 to get y − 2 = − 2 ( x − 1 ) .
Simplify the equation to the standard form, resulting in the equation 2 x + y − 4 = 0 .
Explanation
Understanding the Problem We are given a point ( 1 , 2 ) and a line 2 x + y − 1 = 0 . We need to find the equation of a line that passes through the given point and is parallel to the given line.
Finding the Slope of the Given Line First, let's find the slope of the given line. We can rewrite the equation 2 x + y − 1 = 0 in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. Subtracting 2 x and adding 1 to both sides of the equation, we get y = − 2 x + 1 . Therefore, the slope of the given line is − 2 .
Determining the Slope of the Parallel Line Since the line we want to find is parallel to the given line, it has the same slope. So, the slope of the desired line is also − 2 .
Using the Point-Slope Form Now we can use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is the given point and m is the slope. We have the point ( 1 , 2 ) and the slope − 2 . Plugging these values into the point-slope form, we get y − 2 = − 2 ( x − 1 ) .
Simplifying the Equation Next, we simplify the equation and rewrite it in the standard form A x + B y + C = 0 . Expanding the equation, we have y − 2 = − 2 x + 2 . Adding 2 x to both sides and subtracting 2 from both sides, we get 2 x + y − 4 = 0 .
Final Answer Therefore, the equation of the line that passes through ( 1 , 2 ) and is parallel to the line 2 x + y − 1 = 0 is 2 x + y − 4 = 0 .
Examples
In architecture, when designing parallel structures like hallways or building facades, understanding how to find the equation of a line parallel to another is crucial. For instance, if a designer knows the equation of one wall and needs to create a parallel wall a certain distance away passing through a specific point, they can use the principles of parallel lines and point-slope form to determine the equation of the new wall. This ensures the structures are perfectly aligned and aesthetically pleasing. This concept is also used in urban planning when designing parallel streets or infrastructure.
