To determine which line is parallel to the line [tex]\( 8x + 2y = 12 \)[/tex], we first need to understand that parallel lines share the same slope. Therefore, we will begin by converting the given equation to the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope.
Here are the detailed steps:
1. Start with the given equation:
[tex]\[
8x + 2y = 12
\][/tex]
2. Solve for [tex]\( y \)[/tex]:
[tex]\[
2y = -8x + 12
\][/tex]
[tex]\[
y = -4x + 6
\][/tex]
We rewrite [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] to identify the slope. Here, the slope ([tex]\( m \)[/tex]) is [tex]\( -4 \)[/tex].
3. A line that is parallel to the given line must have the same slope of [tex]\( -4 \)[/tex].
Here are some examples of such lines, each presented in standard form and then converted to slope-intercept form:
### Example 1:
1. Begin with the standard form:
[tex]\[
8x + 2y = 5
\][/tex]
2. Convert it to slope-intercept form:
[tex]\[
2y = -8x + 5
\][/tex]
[tex]\[
y = -4x + \frac{5}{2}
\][/tex]
This line has the same slope of [tex]\( -4 \)[/tex], so it is parallel to the given line.
### Example 2:
1. Begin with the standard form:
[tex]\[
4x + y = 3
\][/tex]
2. Convert it to slope-intercept form:
[tex]\[
y = -4x + 3
\][/tex]
This line also has a slope of [tex]\( -4 \)[/tex], indicating that it is parallel to the given line.
### Example 3:
1. Begin with the standard form:
[tex]\[
12x + 3y = 6
\][/tex]
2. Convert it to slope-intercept form:
[tex]\[
3y = -12x + 6
\][/tex]
[tex]\[
y = -4x + 2
\][/tex]
Again, this line has a slope of [tex]\( -4 \)[/tex], showing that it is parallel to the given line.
Thus, the equations of lines that are parallel to the given line [tex]\( 8x + 2y = 12 \)[/tex] include:
- [tex]\( 8x + 2y = 5 \)[/tex]
- [tex]\( 4x + y = 3 \)[/tex]
- [tex]\( 12x + 3y = 6 \)[/tex]
In summary, any line in the form [tex]\( Ax + By = C \)[/tex] where the slope [tex]\( m = -4 \)[/tex] will be parallel to the given line.
