A given line has the equation $10 x+2 y=-2$. What is the equation, in slope-intercept form, of the line that is parallel to the given line and passes through the point $(0,12)$?
A. $y=-5 x+12$
B. $5 x+y=12$
C. $y-12=5(x-0)$
D. $5 x+y=-1
Answer (2)
Convert the given equation 10 x + 2 y = − 2 to slope-intercept form to find the slope: y = − 5 x − 1 , so the slope is − 5 .
Since parallel lines have the same slope, the new line also has a slope of − 5 .
The line passes through the point ( 0 , 12 ) , which means the y-intercept is 12 .
Write the equation in slope-intercept form: y = − 5 x + 12 .
Explanation
Understanding the Problem We are given the equation of a line 10 x + 2 y = − 2 and asked to find the equation of a line parallel to it that passes through the point ( 0 , 12 ) . The final answer must be in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Finding the Slope of the Given Line First, we need to find the slope of the given line. To do this, we convert the given equation to slope-intercept form. The given equation is 10 x + 2 y = − 2 . We solve for y :
2 y = − 10 x − 2
y = − 5 x − 1
So, the slope of the given line is − 5 .
Determining the Slope and Y-Intercept of the Parallel Line Since parallel lines have the same slope, the slope of the line we are trying to find is also − 5 . We are given that the line passes through the point ( 0 , 12 ) . Since the x-coordinate of this point is 0, this point is the y-intercept. Therefore, b = 12 .
Writing the Equation in Slope-Intercept Form Now we can write the equation of the line in slope-intercept form: y = m x + b . We know that m = − 5 and b = 12 , so the equation is y = − 5 x + 12 .
Final Answer Therefore, the equation of the line that is parallel to the given line and passes through the point ( 0 , 12 ) is y = − 5 x + 12 .
Examples
Imagine you're designing two train tracks that need to run parallel to each other. One track's path is described by the equation 10 x + 2 y = − 2 . You want to lay a new track that runs parallel to the existing one but starts at a different point, say ( 0 , 12 ) on a coordinate plane. By finding the equation of the parallel line, y = − 5 x + 12 , you ensure that the new track runs in the same direction as the original, preventing any collisions or convergences. This concept is also applicable in urban planning, where parallel streets or utility lines need to be designed efficiently.
The equation of the line parallel to 10 x + 2 y = − 2 that passes through ( 0 , 12 ) is y = − 5 x + 12 . Therefore, the correct answer is option A: y = − 5 x + 12 .
;
