Find the equation of the line parallel to $y=4 x+1$ that includes the point ( $-2,-5$ ). Give your answer in Point-Slope Form. $y+5=[?](x-\square)$ Point-Slope Form: $y-y_1=m\left(x_1-x_1\right)$
Answer (1)
The slope of the given line y = 4 x + 1 is 4 .
A line parallel to the given line also has a slope of 4 .
Using the point-slope form y − y 1 = m ( x − x 1 ) with the point ( − 2 , − 5 ) and slope 4 , we get y − ( − 5 ) = 4 ( x − ( − 2 )) .
Simplifying the equation, we have y + 5 = 4 ( x + 2 ) . Therefore, the missing values are 4 and − 2 , so the answer is 4 ; − 2 .
Explanation
Understanding the Problem The problem asks us to find the equation of a line that is parallel to a given line and passes through a specific point. We need to express the equation in point-slope form.
Finding the Slope The given line is y = 4 x + 1 . This is in slope-intercept form, y = m x + b , where m is the slope and b is the y-intercept. Therefore, the slope of the given line is 4 .
Determining the Parallel Slope Since parallel lines have the same slope, the line we are looking for also has a slope of 4 .
Using Point-Slope Form We are given the point ( − 2 , − 5 ) that the line passes through. The point-slope form of a line is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is a point on the line and m is the slope.
Substituting Values Substituting the point ( − 2 , − 5 ) and the slope m = 4 into the point-slope form, we get:
y − ( − 5 ) = 4 ( x − ( − 2 ))
Simplifying, we have:
y + 5 = 4 ( x + 2 )
Final Answer The equation of the line in point-slope form is y + 5 = 4 ( x + 2 ) . Comparing this to the required format y + 5 = [ ?] ( x − □ ) , we can see that the missing values are 4 and − 2 .
Examples
Understanding parallel lines is crucial in various real-world applications, such as architecture and urban planning. For instance, when designing buildings, architects use parallel lines to ensure structural stability and aesthetic appeal. Similarly, city planners use parallel lines to design streets and infrastructure, optimizing traffic flow and minimizing congestion. The concept of parallel lines also extends to navigation, where parallel courses are used to maintain a consistent direction and avoid collisions.
