What is the equation of the line that is parallel to the given line and passes through the point $(-2,2)$?
$y=\frac{1}{5} x+4$
A. $y=\frac{1}{5} x+\frac{12}{5}$
B. $y=-5 x+4$
C. $y=-5 x+\frac{12}{5}$
Answer (2)
Determine the slope of the parallel line: Since the line is parallel to y = 5 1 x + 4 , the slope is 5 1 .
Use the point-slope form: Substitute the slope and the point ( − 2 , 2 ) into y − y 1 = m ( x − x 1 ) to get y − 2 = 5 1 ( x + 2 ) .
Convert to slope-intercept form: Simplify the equation to y = 5 1 x + 5 12 .
State the final answer: The equation of the line is y = 5 1 x + 5 12 .
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. The given line is y = 5 1 x + 4 . Parallel lines have the same slope. Therefore, the line we are looking for has a slope of 5 1 . We also know that the line passes through the point ( − 2 , 2 ) . We can use the point-slope form of a line to find the equation.
Applying Point-Slope Form The point-slope form of a line is given by y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is a point on the line. In our case, m = 5 1 and ( x 1 , y 1 ) = ( − 2 , 2 ) . Substituting these values into the point-slope form, we get:
y − 2 = 5 1 ( x − ( − 2 )) y − 2 = 5 1 ( x + 2 )
Simplifying to Slope-Intercept Form Now, we can simplify the equation to slope-intercept form ( y = m x + b ):
y − 2 = 5 1 x + 5 2 y = 5 1 x + 5 2 + 2 y = 5 1 x + 5 2 + 5 10 y = 5 1 x + 5 12
Final Answer Therefore, the equation of the line that is parallel to y = 5 1 x + 4 and passes through the point ( − 2 , 2 ) is y = 5 1 x + 5 12 .
Examples
Imagine you're designing a ramp for a skateboard park. You want the ramp to have the same slope as another ramp already in the park, ensuring a consistent level of difficulty. If the existing ramp has a slope of 5 1 and you want your ramp to pass through a specific point in the park, say ( − 2 , 2 ) on a coordinate grid, you can use the equation of a line to determine the exact dimensions of your ramp. This ensures that your ramp is parallel to the existing one and meets your desired specifications, making it a fun and challenging addition to the park.
The equation of the line parallel to y = 5 1 x + 4 that passes through the point ( − 2 , 2 ) is y = 5 1 x + 5 12 . This corresponds to option A.
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