Find a general form of an equation of the line through the point [tex]$A$[/tex] that satisfies the given condition.
[tex]$A(-2,5) ; \quad \text { slope } \frac{1}{2}$[/tex]
Answer (2)
Use the point-slope form of a line: y − y 1 = m ( x − x 1 ) .
Substitute the given point A ( − 2 , 5 ) and slope m = 2 1 into the point-slope form: y − 5 = 2 1 ( x + 2 ) .
Simplify the equation: 2 y − 10 = x + 2 .
Convert to the general form: x − 2 y + 12 = 0 .
Explanation
Understanding the Problem We are given a point A ( − 2 , 5 ) and a slope m = 2 1 . We want to find the general form of the equation of the line that passes through point A with the given slope.
Point-Slope Form The point-slope form of a line is given by:
y − y 1 = m ( x − x 1 )
where ( x 1 , y 1 ) is a point on the line and m is the slope of the line.
Substitution and Simplification Substitute the given point A ( − 2 , 5 ) and slope m = 2 1 into the point-slope form:
y − 5 = 2 1 ( x − ( − 2 ))
y − 5 = 2 1 ( x + 2 )
Multiply both sides by 2 to eliminate the fraction:
2 ( y − 5 ) = x + 2
2 y − 10 = x + 2
General Form Rearrange the equation to the general form A x + B y + C = 0 :
x − 2 y + 12 = 0
Thus, the general form of the equation of the line is x − 2 y + 12 = 0 .
Examples
Imagine you are designing a ramp for a skateboard park. You know the ramp needs to pass through a specific point and have a certain slope for safety and usability. Finding the equation of the line that represents the ramp helps you determine the ramp's design and ensure it meets the required specifications. This problem demonstrates how linear equations are used in real-world design and engineering applications.
To find the equation of a line through point A(-2, 5) with a slope of 2 1 , we use the point-slope form to derive x − 2 y + 12 = 0 as the general form. This involves substituting the point and slope into the equation, simplifying, and rearranging it. The resulting equation is in the form A x + B y + C = 0 .
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