The given line passes through the points $(0,-3)$ and $(2, 3)$. What is the equation, in point-slope form, of the line that is parallel to the given line and passes through the point $(-1,-1)$?
A. $y+1=-3(x+1)$
B. $y+1=-\frac{1}{3}(x+1)$
C. $y+1=\frac{1}{3}(x+1)$
D. $y+1=3(x+1)$
Answer (2)
Calculate the slope of the given line using the points ( 0 , − 3 ) and ( 2 , 3 ) : m = 2 − 0 3 − ( − 3 ) = 3 .
Since the line is parallel, it has the same slope: m = 3 .
Use the point-slope form with the point ( − 1 , − 1 ) : y − ( − 1 ) = 3 ( x − ( − 1 )) .
Simplify to get the equation: y + 1 = 3 ( x + 1 ) . The final answer is y + 1 = 3 ( x + 1 ) .
Explanation
Understanding the Problem We are given two points on a line, ( 0 , − 3 ) and ( 2 , 3 ) , and we want to find the equation of a line parallel to this line that passes through the point ( − 1 , − 1 ) . The equation should be in point-slope form.
Finding the Slope First, we need to find the slope of the given line. The slope, m , is given by the formula: m = x 2 − x 1 y 2 − y 1 Using the points ( 0 , − 3 ) and ( 2 , 3 ) , we have: m = 2 − 0 3 − ( − 3 ) = 2 6 = 3 So, the slope of the given line is 3.
Parallel Line Slope Since the line we want to find is parallel to the given line, it will have the same slope. Therefore, the slope of the parallel line is also 3.
Point-Slope Form Now, we use the point-slope form of a line, which is: y − y 1 = m ( x − x 1 ) where m is the slope and ( x 1 , y 1 ) is the point ( − 1 , − 1 ) . Substituting the slope m = 3 and the point ( − 1 , − 1 ) into the point-slope form, we get: y − ( − 1 ) = 3 ( x − ( − 1 )) y + 1 = 3 ( x + 1 )
Final Answer Therefore, the equation of the line in point-slope form is y + 1 = 3 ( x + 1 ) .
Examples
Imagine you're designing a ramp for a skateboard park. You know the slope you want for the ramp, and you need it to pass through a specific point. Using the point-slope form, you can easily determine the equation of the ramp. This ensures the ramp has the correct steepness and starts at the desired location. The point-slope form is also useful in physics, for example, when determining the trajectory of a projectile or analyzing linear motion with a constant slope.
To find the equation of a line parallel to the one through the points (0, -3) and (2, 3) that passes through (-1, -1), we first calculate the slope of the original line, which is 3. Using this slope and the point (-1, -1) in the point-slope form, we find the equation to be y + 1 = 3 ( x + 1 ) . Therefore, the correct answer is option D.
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