A given line has the equation [tex]\(10x + 2y = -2\)[/tex].

What is the equation, in slope-intercept form, of the line that is parallel to the given line and passes through the point [tex]\((0,12)\)[/tex]?

A. [tex]\(y = -5x + 12\)[/tex]

B. [tex]\(5x + y = 12\)[/tex]

C. [tex]\(y - 12 = 5(x - 0)\)[/tex]

D. [tex]\(5x + y = -1\)[/tex]



Answer :

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]