To determine a line that is parallel to the given line [tex]\( 8x + 2y = 12 \)[/tex], we need to follow these steps:
1. Rewrite the equation in slope-intercept form:
The standard form of a line is [tex]\( Ax + By = C \)[/tex]. The slope-intercept form is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope, and [tex]\( b \)[/tex] represents the y-intercept.
Starting with the given equation:
[tex]\[
8x + 2y = 12
\][/tex]
We want to solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
2y = -8x + 12
\][/tex]
Next, divide every term by 2:
[tex]\[
y = -4x + 6
\][/tex]
From this, we see that the equation is now in the form [tex]\( y = mx + b \)[/tex], where the slope [tex]\( m \)[/tex] is [tex]\(-4\)[/tex].
2. Identify the slope for any parallel line:
For two lines to be parallel, they must have the same slope. Therefore, any line parallel to [tex]\( 8x + 2y = 12 \)[/tex] will also have a slope of [tex]\(-4\)[/tex].
3. Construct the equation of a parallel line:
To write an equation for a line parallel to [tex]\( 8x + 2y = 12 \)[/tex], we use the slope [tex]\(-4\)[/tex] but can choose a different y-intercept [tex]\( b \)[/tex]. The general form of such an equation would be:
[tex]\[
y = -4x + b
\][/tex]
Where [tex]\( b \)[/tex] can be any real number, representing the y-intercept of the parallel line.
Example:
Let's take [tex]\( b = 3 \)[/tex].
The equation of the parallel line would then be:
[tex]\[
y = -4x + 3
\][/tex]
Thus, the line [tex]\( y = -4x + 3 \)[/tex] is one example of a line that is parallel to the original line [tex]\( 8x + 2y = 12 \)[/tex].
