To find the equation of a line that is parallel to the given line and passes through a specific point, we need to follow these steps:
1. Determine the slope of the given line:
The given line equation is [tex]\(10x + 2y = -2\)[/tex]. To find the slope, we first need to convert this equation to slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope.
Start by isolating [tex]\(y\)[/tex]:
[tex]\[
10x + 2y = -2
\][/tex]
Subtract [tex]\(10x\)[/tex] from both sides:
[tex]\[
2y = -10x - 2
\][/tex]
Divide every term by 2:
[tex]\[
y = -5x - 1
\][/tex]
From this form, it is clear that the slope [tex]\(m\)[/tex] of the given line is [tex]\(-5\)[/tex].
2. Use the slope of the parallel line:
Since parallel lines share the same slope, the slope of the new line must also be [tex]\(-5\)[/tex].
3. Find the equation using the point-slope form:
We need the equation of the line that passes through the point [tex]\((0,12)\)[/tex] and has a slope of [tex]\(-5\)[/tex]. The point-slope form of a line equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\((x_1, y_1) = (0, 12)\)[/tex] and [tex]\(m = -5\)[/tex].
Substitute the given point and the slope into the point-slope form equation:
[tex]\[
y - 12 = -5(x - 0)
\][/tex]
4. Simplify to slope-intercept form:
Simplify the equation:
[tex]\[
y - 12 = -5x
\][/tex]
Add 12 to both sides:
[tex]\[
y = -5x + 12
\][/tex]
Therefore, 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]
Thus, the correct option is:
[tex]\[
y = -5x + 12
\][/tex]
