Question
what is the equation of the line that passes through (-2,-5) and is parallel to y=x+3
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GPT-4o mini
To find the equation of a line that passes through the point \((-2, -5)\) and is parallel to the line given by the equation \(y = x + 3\), we first note that parallel lines have the same slope.
The slope of the line \(y = x + 3\) is 1 (since it can be written in the slope-intercept form \(y = mx + b\), where \(m\) is the slope).
Since the new line is parallel, it also has a slope \(m = 1\).
Now, we can use the point-slope form of the equation of a line, which 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. Plugging in the point \((-2, -5)\) and the slope \(1\):
\[
y - (-5) = 1(x - (-2))
\]
Simplifying this:
\[
y + 5 = 1(x + 2)
\]
\[
y + 5 = x + 2
\]
\[
y = x + 2 - 5
\]
\[
y = x - 3
\]
Thus, the equation of the line that passes through the point \((-2, -5)\) and is parallel to the line \(y = x + 3\) is:
\[
y = x - 3
\]
The slope of the line \(y = x + 3\) is 1 (since it can be written in the slope-intercept form \(y = mx + b\), where \(m\) is the slope).
Since the new line is parallel, it also has a slope \(m = 1\).
Now, we can use the point-slope form of the equation of a line, which 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. Plugging in the point \((-2, -5)\) and the slope \(1\):
\[
y - (-5) = 1(x - (-2))
\]
Simplifying this:
\[
y + 5 = 1(x + 2)
\]
\[
y + 5 = x + 2
\]
\[
y = x + 2 - 5
\]
\[
y = x - 3
\]
Thus, the equation of the line that passes through the point \((-2, -5)\) and is parallel to the line \(y = x + 3\) is:
\[
y = x - 3
\]
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