Certainly! Let's go step-by-step to determine the equation of a line parallel to the given line [tex]\( 3x + 2y = 8 \)[/tex] and passing through the point [tex]\((-2, 5)\)[/tex].
1. Identify the slope of the given line:
- Start by converting the given equation to slope-intercept form [tex]\( y = mx + b \)[/tex].
- The given equation is [tex]\( 3x + 2y = 8 \)[/tex].
- Rearrange it to isolate [tex]\( y \)[/tex]:
[tex]\[
2y = -3x + 8
\][/tex]
[tex]\[
y = -\frac{3}{2}x + 4
\][/tex]
- The slope [tex]\( m \)[/tex] of the given line is [tex]\(-\frac{3}{2}\)[/tex].
2. Use the slope to find the equation of the parallel line:
- Parallel lines have the same slope. Thus, the slope of our new line will also be [tex]\(-\frac{3}{2}\)[/tex].
- Use the point-slope form of the equation of a line, which is [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Here, [tex]\( (x_1, y_1) \)[/tex] is the point [tex]\((-2, 5)\)[/tex]
- Slope [tex]\( m = -\frac{3}{2} \)[/tex]
- Substitute the given point and slope into the point-slope form:
[tex]\[
y - 5 = -\frac{3}{2}(x + 2)
\][/tex]
3. Convert this equation to slope-intercept form ( [tex]\( y = mx + b \)[/tex] ):
- Begin by distributing the slope on the right-hand side:
[tex]\[
y - 5 = -\frac{3}{2}x - 3
\][/tex]
- Add 5 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{3}{2}x - 3 + 5
\][/tex]
[tex]\[
y = -\frac{3}{2}x + 2
\][/tex]
So, the equation of the line parallel to [tex]\( 3x + 2y = 8 \)[/tex] that passes through [tex]\((-2, 5)\)[/tex] is [tex]\( y = -\frac{3}{2}x + 2 \)[/tex].
Thus, the correct answers to be selected in the drop-down menus are:
- The first [tex]\(\square\)[/tex] should be filled with "[tex]\(-1.5x\)[/tex]" or "[tex]\(-\frac{3}{2}x\)[/tex]".
- The second [tex]\(\square\)[/tex] should be filled with "[tex]\(+ 2.0\)[/tex]".
In other words, fill in the blanks as:
[tex]\[ y = -1.5x + 2.0 \][/tex]
