Question
Calvin states that it is not possible to gind the equation of a line parallel to y=2 and passing through the point (5,-1) using the point slope form because the equation of the line does not have an x-term. Explain the error in Calvin’s reasoning. Then find the equation of the parallel line.
Answers
GPT-4o mini
Calvin's reasoning is flawed because the point-slope form of a line can still be used to find the equation of a line that is parallel to another line, even if the other line does not have an \(x\)-term. The equation \(y = 2\) represents a horizontal line that has a slope of 0. Lines that are parallel to each other share the same slope. Therefore, the line parallel to \(y = 2\) will also have a slope of 0, meaning it will also be a horizontal line.
Since the desired line is parallel to \(y = 2\) and needs to pass through the point \((5, -1)\), we know that the line should have the form \(y = k\), where \(k\) is a constant. The line will have the same \(y\)-value across all \(x\)-values.
To find the equation of the line parallel to \(y = 2\) that passes through the point \((5, -1)\), we simply take the \(y\)-coordinate of the point. Therefore, the equation of the new line is:
\[
y = -1
\]
This line is horizontal and parallel to \(y = 2\).
Since the desired line is parallel to \(y = 2\) and needs to pass through the point \((5, -1)\), we know that the line should have the form \(y = k\), where \(k\) is a constant. The line will have the same \(y\)-value across all \(x\)-values.
To find the equation of the line parallel to \(y = 2\) that passes through the point \((5, -1)\), we simply take the \(y\)-coordinate of the point. Therefore, the equation of the new line is:
\[
y = -1
\]
This line is horizontal and parallel to \(y = 2\).