To find the equation of a line that is parallel to a given line and passes through a specific point, you need to follow these steps:
### Step 1: Convert the Given Line to Slope-Intercept Form
The given line equation is:
[tex]\[ 10x + 2y = -2 \][/tex]
First, we need to convert this equation into the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
1. Isolate \( y \) on one side of the equation:
[tex]\[ 2y = -10x - 2 \][/tex]
2. Divide all terms by 2:
[tex]\[ y = -5x - 1 \][/tex]
Now we have the given line in slope-intercept form \( y = -5x - 1 \). The slope (\( m \)) of the given line is \( -5 \).
### Step 2: Use the Slope of the Parallel Line
A line that is parallel to the given line will have the same slope. Therefore, the slope of the new line will also be \( -5 \).
### Step 3: Use the Point-Slope Form of the Equation
The new line passes through the point \( (0, 12) \). To find the equation of the line, we can use the point-slope form of a line's equation:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substitute \( (x_1, y_1) = (0, 12) \) and \( m = -5 \):
[tex]\[ y - 12 = -5(x - 0) \][/tex]
Simplify the equation:
[tex]\[ y - 12 = -5x \][/tex]
[tex]\[ y = -5x + 12 \][/tex]
### Conclusion
The equation of the line that is parallel to \( 10x + 2y = -2 \) and passes through the point \( (0, 12) \) is:
[tex]\[ y = -5x + 12 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{y = -5x + 12} \][/tex]
