To find the equation of the line containing the points \((-4, 7)\) and \((2, -2)\), we will follow these steps:
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Calculate the slope (m): The slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points \((-4, 7)\) and \((2, -2)\):
- \((x_1, y_1) = (-4, 7)\)
- \((x_2, y_2) = (2, -2)\)
Now substitute these values into the slope formula:
\[ m = \frac{-2 - 7}{2 - (-4)} = \frac{-9}{2 + 4} = \frac{-9}{6} = -\frac{3}{2} \]
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Use the slope-intercept form: The slope-intercept form of a line is given by:
\[ y = mx + b \]
We know \(m = -\frac{3}{2}\). We'll use one of the points to find \(b\). Let's use the point \((-4, 7)\).
Substitute \(x = -4\), \(y = 7\), and \(m = -\frac{3}{2}\) into the equation:
\[ 7 = -\frac{3}{2}(-4) + b \]
Calculate \(-\frac{3}{2}(-4)\):
\[ 7 = 6 + b \]
Now, solve for \(b\):
\[ b = 7 - 6 = 1 \]
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Write the final equation: Now that we have the slope \(m = -\frac{3}{2}\) and \(b = 1\), we can write the equation of the line:
\[ y = -\frac{3}{2}x + 1 \]
So, the equation of the line containing the points \((-4, 7)\) and \((2, -2)\) is:
\[ y = -\frac{3}{2}x + 1 \]