What is the equation, in point-slope form, of the line that is parallel to the given line and passes through the point $(-3,1)$?
$y-1=-\frac{3}{2}(x+3)$
$y-1=-\frac{2}{3}(x+3)$
$y-1=\frac{2}{3}(x+3)$
$y-1=\frac{3}{2}(x+3)$
Answer (1)
The problem asks for the equation of a line in point-slope form that is parallel to a given line and passes through the point ( − 3 , 1 ) .
The point-slope form is y − y 1 = m ( x − x 1 ) .
The line passes through ( − 3 , 1 ) , so the equation is y − 1 = m ( x + 3 ) .
Without the original equation, we cannot determine the slope m .
We cannot determine the correct answer without the original equation. Therefore, we cannot provide a final boxed answer.
Explanation
Understanding the Problem We are given a point ( − 3 , 1 ) and asked to find the equation of a line in point-slope form that is parallel to a given line. 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. Parallel lines have the same slope. We need to identify the correct equation from the given options.
Analyzing the Point-Slope Form The given options are in point-slope form, y − y 1 = m ( x − x 1 ) . We know that the line passes through the point ( − 3 , 1 ) , so x 1 = − 3 and y 1 = 1 . Thus, the equation must be of the form y − 1 = m ( x − ( − 3 )) , which simplifies to y − 1 = m ( x + 3 ) . We need to determine the correct slope m from the given options.
Identifying the Correct Slope The options provide different slopes. Since we are looking for a line parallel to a given line, we need to identify which of the given options represents a line in point-slope form. The options are:
y − 1 = − 2 3 ( x + 3 )
y − 1 = − 3 2 ( x + 3 )
y − 1 = 3 2 ( x + 3 )
y − 1 = 2 3 ( x + 3 )
Choosing the Correct Option Without knowing the original line, we cannot determine the exact slope of the parallel line. However, we can analyze the given options to see which one is in the correct point-slope form with the given point ( − 3 , 1 ) . All options are in the form y − 1 = m ( x + 3 ) , so we need to choose the correct slope m from the options. Since we don't have any other information, we must assume one of the options is the correct answer.
Final Answer Since we don't have the original equation, we can't determine the slope. However, we can see that all options are in point-slope form using the point (-3, 1). Without more information, we cannot determine which slope is correct. However, if we assume that one of the options is the correct answer, we can look at the options.
Reiterating the Issue Since we don't have the equation of the original line, we cannot determine the slope of the parallel line. However, we can analyze the given options to see which one is in the correct point-slope form with the given point ( − 3 , 1 ) . All options are in the form y − 1 = m ( x + 3 ) , so we need to choose the correct slope m from the options. Without more information, we must assume one of the options is the correct answer. The problem states that the line is parallel to the given line. However, the given line is not provided. Therefore, we cannot determine the correct answer.
Examples
Understanding point-slope form is crucial in various real-world applications. For instance, imagine you are tracking the altitude of a hot air balloon. If you know the balloon's altitude at one point in time and its rate of ascent (slope), you can use the point-slope form to predict its altitude at any other time. Similarly, in economics, if you know the cost of producing a certain number of items and the variable cost per item, you can use the point-slope form to model the total cost of production. These examples highlight how the point-slope form helps in making predictions and modeling linear relationships in practical scenarios.
