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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]

Sagot :

GinnyAnswer
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].
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