To find the equation of a line parallel to the given line [tex]\(10x + 2y = -2\)[/tex] and passing through the point [tex]\((0, 12)\)[/tex], follow these steps:
1. Rewrite the given equation in 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. Start by isolating [tex]\(y\)[/tex] in the given equation.
[tex]\(10x + 2y = -2\)[/tex]
Subtract [tex]\(10x\)[/tex] from both sides:
[tex]\(2y = -10x - 2\)[/tex]
Divide everything by 2:
[tex]\(y = -5x - 1\)[/tex]
So, the slope [tex]\(m\)[/tex] of the given line is [tex]\(-5\)[/tex].
2. Find the slope of the parallel line:
Parallel lines have the same slope. Therefore, the slope of the line we are looking for is also [tex]\(-5\)[/tex].
3. Use the point-slope form to find the equation of the parallel line:
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 and [tex]\(m\)[/tex] is the slope. Here, the point is [tex]\((0, 12)\)[/tex] and the slope is [tex]\(-5\)[/tex].
Substitute the values into the point-slope form:
[tex]\(y - 12 = -5(x - 0)\)[/tex]
Simplify the equation:
[tex]\(y - 12 = -5x\)[/tex]
4. Rewrite the equation in slope-intercept form:
Add 12 to both sides to get [tex]\(y\)[/tex] by itself:
[tex]\(y = -5x + 12\)[/tex]
Thus, the equation of the line parallel to [tex]\(10x + 2y = -2\)[/tex] and passing through the point [tex]\((0, 12)\)[/tex] in slope-intercept form is:
[tex]\[ y = -5x + 12 \][/tex]
The correct answer is [tex]\( y = -5x + 12 \)[/tex].
