Find the equation of the line with slope = [tex]$\frac{1}{3}$[/tex] and passing through the point (4,7).
Answer (1)
Use the point-slope form of a line: y − y 1 = m ( x − x 1 ) .
Substitute the given slope m = 3 1 and point ( 4 , 7 ) into the equation: y − 7 = 3 1 ( x − 4 ) .
Simplify the equation to slope-intercept form: y = 3 1 x − 3 4 + 7 .
Combine constants to get the final equation: y = 3 1 x + 3 17 .
y = 3 1 x + 3 17
Explanation
Understanding the Problem We are given the slope of a line, which is 3 1 , and a point that the line passes through, which is ( 4 , 7 ) . Our goal is to find the equation of this line.
Using Point-Slope Form We will use the point-slope form of a linear equation, which is given by: y − y 1 = m ( x − x 1 ) where m is the slope of the line and ( x 1 , y 1 ) is a point on the line.
Substituting Values We are given that the slope m = 3 1 and the point ( x 1 , y 1 ) = ( 4 , 7 ) . Substituting these values into the point-slope form, we get: y − 7 = 3 1 ( x − 4 )
Distributing Now, we simplify the equation to get it into slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. First, distribute the 3 1 on the right side: y − 7 = 3 1 x − 3 4
Isolating y Next, add 7 to both sides of the equation to isolate y :
y = 3 1 x − 3 4 + 7
Simplifying To combine the constants, we need a common denominator. Since 7 = 3 21 , we have: y = 3 1 x − 3 4 + 3 21 y = 3 1 x + 3 17
Final Answer So, the equation of the line in slope-intercept form is: y = 3 1 x + 3 17
Examples
Imagine you're designing a ramp for a skateboard park. You know the ramp needs to pass through a certain point and have a specific slope for safety and usability. Finding the equation of the line that represents the ramp's surface helps you determine the exact dimensions and ensure a smooth transition for skateboarders. This blend of algebra and practical design ensures the ramp meets the required specifications, making it both fun and safe to use.
