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Part 1
If the slope of a line and a point on the line are​ known, the equation of the line can be found using the​ slope-intercept form, yequalsmxplusb. To do​ so, substitute the value of the slope and the values of x and y using the coordinates of the given​ point, then determine the value of b.
Using the above​ technique, find the equation of the line containing the points ​(minus4​,7​) and left parenthesis 2 comma negative 2 right parenthesis.

1 answer

To find the equation of the line containing the points \((-4, 7)\) and \((2, -2)\), we will follow these steps:

  1. 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:

    \[ m = \frac{-2 - 7}{2 - (-4)} = \frac{-9}{2 + 4} = \frac{-9}{6} = -\frac{3}{2} \]

  2. 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 \(b\):

    \[ b = 7 - 6 = 1 \]

  3. 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:

\[ y = -\frac{3}{2}x + 1 \]