To find the equation of the line that is parallel to the given line [tex]\(10x + 2y = -2\)[/tex] and passes through the point [tex]\((0, 12)\)[/tex], we will follow these steps:
### Step 1: Convert the Given Line to Slope-Intercept Form
The slope-intercept form of a line is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
First, we need to convert the given line [tex]\(10x + 2y = -2\)[/tex] to this form.
1. Solve for [tex]\(y\)[/tex]:
[tex]\[
2y = -10x - 2
\][/tex]
2. Divide both sides by 2:
[tex]\[
y = -5x - 1
\][/tex]
Therefore, the slope of the given line [tex]\(m\)[/tex] is [tex]\(-5\)[/tex].
### Step 2: Use the Slope of the Parallel Line
Parallel lines have the same slope. Therefore, the line we are looking for also has a slope of [tex]\(-5\)[/tex].
### Step 3: Use Point-Slope Form to Find the Equation
We need to find the equation of the line with slope [tex]\(-5\)[/tex] that passes through the point [tex]\((0, 12)\)[/tex].
The point-slope form of a line is [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\((x_1, y_1)\)[/tex] is a point on the line.
1. Plug in the slope [tex]\(m = -5\)[/tex] and the point [tex]\((0, 12)\)[/tex]:
[tex]\[
y - 12 = -5(x - 0)
\][/tex]
2. Simplify the equation:
[tex]\[
y - 12 = -5x
\][/tex]
[tex]\[
y = -5x + 12
\][/tex]
### Conclusion
The equation of the line that is parallel to [tex]\(10x + 2y = -2\)[/tex] and passes through the point [tex]\((0, 12)\)[/tex] is [tex]\(y = -5x + 12\)[/tex].
So the correct choice from the given options is:
[tex]\[
y = -5x + 12
\][/tex]
