To find a line parallel to the given line [tex]\(8x + 2y = 12\)[/tex], we need to follow these steps:
1. Convert the given line to slope-intercept form [tex]\( y = mx + b \)[/tex]:
Start with the given equation:
[tex]\[
8x + 2y = 12
\][/tex]
To convert this into slope-intercept form, [tex]\( y = mx + b \)[/tex], we solve for [tex]\( y \)[/tex].
First, isolate the [tex]\( y \)[/tex]-term:
[tex]\[
2y = -8x + 12
\][/tex]
Next, divide every term by 2 to solve for [tex]\( y \)[/tex]:
[tex]\[
y = -4x + 6
\][/tex]
2. Identify the slope [tex]\( m \)[/tex]:
The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. From the rearranged equation [tex]\( y = -4x + 6 \)[/tex], we see that the slope [tex]\( m \)[/tex] is [tex]\(-4\)[/tex].
3. Form the equation of a line parallel to the given line:
Lines that are parallel share the same slope. Therefore, any line parallel to the given line must also have a slope of [tex]\(-4\)[/tex].
The general form of a line with slope [tex]\(-4\)[/tex] is:
[tex]\[
y = -4x + b
\][/tex]
where [tex]\( b \)[/tex] can be any real number.
4. Express the parallel line in the standard form:
To express the equation of the parallel line in the same format as the given line, we use:
[tex]\[
y = -4x + b
\][/tex]
This can be rewritten to:
[tex]\[
y = -4x + b
\][/tex]
Therefore, the equation of a line parallel to [tex]\( 8x + 2y = 12 \)[/tex] is:
[tex]\[
\boxed{y = -4x + b}
\][/tex]
