What is the equation of the line that is parallel to the line $5x + 2y = 12$ and passes through the point $(-2, 4)$?
A. $y=-\frac{5}{2}x-1$
B. $y=-\frac{5}{2}x+5$
C. $y=\frac{2}{5}x-1$
D. $y=\frac{2}{5}x+5$
Answer (1)
Find the slope of the given line 5 x + 2 y = 12 by rewriting it in slope-intercept form: y = − 2 5 x + 6 . The slope is − 2 5 .
Since the desired line is parallel, it has the same slope: m = − 2 5 .
Use the point-slope form with the point ( − 2 , 4 ) : y − 4 = − 2 5 ( x + 2 ) .
Convert to slope-intercept form: y = − 2 5 x − 1 . The equation of the line is y = − 2 5 x − 1 .
Explanation
Understanding the Problem We are given a line 5 x + 2 y = 12 and a point ( − 2 , 4 ) . We need to find the equation of a line that is parallel to the given line and passes through the given point.
Finding the Slope of the Given Line First, let's find the slope of the given line. To do this, we can rewrite the equation in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. Starting with 5 x + 2 y = 12 , we can solve for y :
2 y = − 5 x + 12
y = − 2 5 x + 6
So, the slope of the given line is − 2 5 .
Determining the Slope of the Parallel Line Since the line we are looking for is parallel to the given line, it has the same slope. Therefore, the slope of the line we want to find is also − 2 5 .
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 m is the slope and ( x 1 , y 1 ) is the given point. In this case, m = − 2 5 and ( x 1 , y 1 ) = ( − 2 , 4 ) . Plugging these values into the point-slope form, we get:
y − 4 = − 2 5 ( x − ( − 2 ))
y − 4 = − 2 5 ( x + 2 )
Converting to Slope-Intercept Form Now, let's rewrite the equation in slope-intercept form, y = m x + b :
y − 4 = − 2 5 x − 5
y = − 2 5 x − 5 + 4
y = − 2 5 x − 1
Final Answer Therefore, the equation of the line that is parallel to the line 5 x + 2 y = 12 and passes through the point ( − 2 , 4 ) is y = − 2 5 x − 1 .
Examples
Understanding parallel lines is crucial in various real-world applications. For instance, consider city planning where streets are often designed to be parallel to each other. If a new street needs to be constructed parallel to an existing one, and it must pass through a specific location, the principles used in this problem can be applied. Similarly, in architecture, parallel lines are fundamental in building design, ensuring structural stability and aesthetic appeal. Knowing how to determine the equation of a parallel line helps in accurately planning and executing such projects.
