Which equation represents a line that passes through $(-2,4)$ and has a slope of $\frac{2}{5}$?
A. $y-4=\frac{2}{5}(x+2)$
B. $y+4=\frac{2}{5}(x-2)$
C. $y+2=\frac{2}{5}(x-4)$
D. $y-2=\frac{2}{5}(x+4)$
Answer (2)
Use the point-slope form of a linear equation: y − y 1 = m ( x − x 1 ) .
Substitute the given point ( − 2 , 4 ) and slope 5 2 into the formula: y − 4 = 5 2 ( x − ( − 2 )) .
Simplify the equation: y − 4 = 5 2 ( x + 2 ) .
The equation representing the line is: y − 4 = 5 2 ( x + 2 ) .
Explanation
Understanding the Problem We are given a point ( − 2 , 4 ) and a slope m = 5 2 . We need to find the equation of the line that passes through this point and has this slope.
Using Point-Slope Form The point-slope form of a linear equation 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.
Substituting Values Substitute the given point ( − 2 , 4 ) for ( x 1 , y 1 ) and the slope 5 2 for m into the point-slope form: y − 4 = 5 2 ( x − ( − 2 )) y − 4 = 5 2 ( x + 2 )
Finding the Matching Equation Comparing the derived equation with the given options, we find that the equation that matches is: y − 4 = 5 2 ( x + 2 )
Final Answer Therefore, the equation of the line that passes through ( − 2 , 4 ) and has a slope of 5 2 is y − 4 = 5 2 ( x + 2 ) .
Examples
Understanding linear equations is crucial in many real-world applications. For instance, if you're tracking the distance a car travels over time at a constant speed, you're essentially dealing with a linear relationship. If a car starts 2 miles away from your home and travels at a speed of 5 2 miles per minute, the equation y − 2 = 5 2 x can represent its position y relative to your home after x minutes. This concept extends to various fields like physics, economics, and computer science, making it a fundamental tool for modeling and predicting outcomes.
The equation representing the line that passes through (-2, 4) and has a slope of 5 2 is y − 4 = 5 2 ( x + 2 ) . Therefore, the correct answer is Option A. This form indicates that the specified conditions of the line are satisfied.
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