Question

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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.

Answer 1

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:

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