To find the equation of the line that is parallel to [tex]\(3x + 2y = 8\)[/tex] and passes through the point [tex]\((-2, 5)\)[/tex], let's follow these steps:
1. Determine the slope of the given line:
The equation [tex]\(3x + 2y = 8\)[/tex] is in standard form. We first need to convert it to the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
[tex]\[ 3x + 2y = 8 \][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[ 2y = -3x + 8 \][/tex]
[tex]\[ y = -\frac{3}{2}x + 4 \][/tex]
Therefore, the slope of the given line is [tex]\(m = -\frac{3}{2}\)[/tex].
2. Find the slope of the parallel line:
Parallel lines have the same slope. So, the slope of the new line will also be [tex]\(m = -\frac{3}{2}\)[/tex].
3. Use the point-slope form to find the equation of the new line:
We now use the point-slope form of the equation of a line, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\((x_1, y_1) = (-2, 5)\)[/tex], and [tex]\(m = -\frac{3}{2}\)[/tex]:
[tex]\[ y - 5 = -\frac{3}{2}(x - (-2)) \][/tex]
[tex]\[ y - 5 = -\frac{3}{2}(x + 2) \][/tex]
4. Simplify the equation:
Distribute [tex]\(-\frac{3}{2}\)[/tex]:
[tex]\[ y - 5 = -\frac{3}{2}x - 3 \][/tex]
Add 5 to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[ y = -\frac{3}{2}x - 3 + 5 \][/tex]
[tex]\[ y = -\frac{3}{2}x + 2 \][/tex]
Therefore, the equation of the line parallel to [tex]\(3x + 2y = 8\)[/tex] and passing through the point [tex]\((-2, 5)\)[/tex] is [tex]\(y = -\frac{3}{2}x + 2\)[/tex].
So, the correct selections for the drop-down menus are:
[tex]\[ y = -\frac{3}{2} \, \text{x} + 2 \][/tex]
