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write an equation of a line passing through the points (-5,6) and (5,2)
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write an equation of a line passing through the points (-5,6) and (5,2) in slope intercept form
Answered by
GPT-4o mini
To find the equation of the line passing through the points \((-5, 6)\) and \((5, 2)\), we first need to determine the slope (m) of the line. The slope can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
For the points \((-5, 6)\) (where \(x_1 = -5\) and \(y_1 = 6\)) and \((5, 2)\) (where \(x_2 = 5\) and \(y_2 = 2\)), we can substitute these coordinates into the formula:
\[
m = \frac{2 - 6}{5 - (-5)} = \frac{2 - 6}{5 + 5} = \frac{-4}{10} = -\frac{2}{5}
\]
Now that we have the slope \(m = -\frac{2}{5}\), we can use the point-slope form of the equation of a line, which is:
\[
y - y_1 = m(x - x_1)
\]
We can use either of the two points; let's use \((-5, 6)\):
\[
y - 6 = -\frac{2}{5}(x - (-5))
\]
This simplifies to:
\[
y - 6 = -\frac{2}{5}(x + 5)
\]
Now, distribute the slope:
\[
y - 6 = -\frac{2}{5}x - 2
\]
Now, add 6 to both sides:
\[
y = -\frac{2}{5}x - 2 + 6
\]
This simplifies to:
\[
y = -\frac{2}{5}x + 4
\]
Thus, the equation of the line in slope-intercept form is:
\[
\boxed{y = -\frac{2}{5}x + 4}
\]
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
For the points \((-5, 6)\) (where \(x_1 = -5\) and \(y_1 = 6\)) and \((5, 2)\) (where \(x_2 = 5\) and \(y_2 = 2\)), we can substitute these coordinates into the formula:
\[
m = \frac{2 - 6}{5 - (-5)} = \frac{2 - 6}{5 + 5} = \frac{-4}{10} = -\frac{2}{5}
\]
Now that we have the slope \(m = -\frac{2}{5}\), we can use the point-slope form of the equation of a line, which is:
\[
y - y_1 = m(x - x_1)
\]
We can use either of the two points; let's use \((-5, 6)\):
\[
y - 6 = -\frac{2}{5}(x - (-5))
\]
This simplifies to:
\[
y - 6 = -\frac{2}{5}(x + 5)
\]
Now, distribute the slope:
\[
y - 6 = -\frac{2}{5}x - 2
\]
Now, add 6 to both sides:
\[
y = -\frac{2}{5}x - 2 + 6
\]
This simplifies to:
\[
y = -\frac{2}{5}x + 4
\]
Thus, the equation of the line in slope-intercept form is:
\[
\boxed{y = -\frac{2}{5}x + 4}
\]
Answered by
GPT-4o mini
To find the equation of the line that passes through the points \((-5, 6)\) and \((5, 2)\) in slope-intercept form (\(y = mx + b\)), we first need to determine the slope \(m\) using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
For the points \((-5, 6)\) (where \(x_1 = -5\) and \(y_1 = 6\)) and \((5, 2)\) (where \(x_2 = 5\) and \(y_2 = 2\)), we substitute into the formula:
\[
m = \frac{2 - 6}{5 - (-5)} = \frac{2 - 6}{5 + 5} = \frac{-4}{10} = -\frac{2}{5}
\]
Now that we have the slope \(m = -\frac{2}{5}\), we can use the slope-intercept form, which is \(y = mx + b\). To find \(b\), the y-intercept, we can substitute one of the points into the equation. Let's use the point \((-5, 6)\):
\[
6 = -\frac{2}{5}(-5) + b
\]
Calculating \(-\frac{2}{5}(-5)\):
\[
6 = 2 + b
\]
Next, we solve for \(b\):
\[
b = 6 - 2 = 4
\]
Now we can write the equation of the line in slope-intercept form:
\[
y = -\frac{2}{5}x + 4
\]
Thus, the equation of the line passing through the points \((-5, 6)\) and \((5, 2)\) is:
\[
\boxed{y = -\frac{2}{5}x + 4}
\]
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
For the points \((-5, 6)\) (where \(x_1 = -5\) and \(y_1 = 6\)) and \((5, 2)\) (where \(x_2 = 5\) and \(y_2 = 2\)), we substitute into the formula:
\[
m = \frac{2 - 6}{5 - (-5)} = \frac{2 - 6}{5 + 5} = \frac{-4}{10} = -\frac{2}{5}
\]
Now that we have the slope \(m = -\frac{2}{5}\), we can use the slope-intercept form, which is \(y = mx + b\). To find \(b\), the y-intercept, we can substitute one of the points into the equation. Let's use the point \((-5, 6)\):
\[
6 = -\frac{2}{5}(-5) + b
\]
Calculating \(-\frac{2}{5}(-5)\):
\[
6 = 2 + b
\]
Next, we solve for \(b\):
\[
b = 6 - 2 = 4
\]
Now we can write the equation of the line in slope-intercept form:
\[
y = -\frac{2}{5}x + 4
\]
Thus, the equation of the line passing through the points \((-5, 6)\) and \((5, 2)\) is:
\[
\boxed{y = -\frac{2}{5}x + 4}
\]
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