Question

find\:the\:equation\:in\:slope\:intercept\:form\:for\:the\:line\:that\:goes\:through\:the\:point\:\left(-4,\:5\right)\:and\:is\:parallel\:to\:the\:line\:that\:passes\:through\:\left(-4,\:0\right)\:and\:\left(4,\:-2\right)

Answer 1

To find the equation of the line that goes through the point \((-4, 5)\) and is parallel to the line passing through the points \((-4, 0)\) and \((4, -2)\), we need to follow these steps:

### Step 1: Find the slope of the line passing through \((-4, 0)\) and \((4, -2)\).

The formula for finding the slope \(m\) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the given points:

\[
m = \frac{-2 - 0}{4 - (-4)} = \frac{-2}{4 + 4} = \frac{-2}{8} = -\frac{1}{4}
\]

### Step 2: Use the slope to find the equation of the line parallel to it.

Since the line we want to find is parallel to the original line, it will have the same slope, which is \(-\frac{1}{4}\).

### Step 3: Use the point-slope form of the equation of a line to find the equation.

The point-slope form is given by:

\[
y - y_1 = m(x - x_1)
\]

Substituting the slope \(-\frac{1}{4}\) and the point \((-4, 5)\):

\[
y - 5 = -\frac{1}{4}(x - (-4))
\]

This simplifies to:

\[
y - 5 = -\frac{1}{4}(x + 4)
\]

### Step 4: Distribute and solve for \(y\).

Distributing the slope on the right-hand side:

\[
y - 5 = -\frac{1}{4}x - 1
\]

Now, add 5 to both sides:

\[
y = -\frac{1}{4}x - 1 + 5
\]
\[
y = -\frac{1}{4}x + 4
\]

### Step 5: Write the equation in slope-intercept form \(y = mx + b\).

The final equation of the line in slope-intercept form is:

\[
\boxed{y = -\frac{1}{4}x + 4}
\]