To find the equation of the line that is parallel to [tex]\(5x + 2y = 12\)[/tex] and passes through the point [tex]\((-2, 4)\)[/tex], follow these steps:
1. Find the slope of the original line:
The given line is [tex]\(5x + 2y = 12\)[/tex]. To determine its slope, rewrite the line equation in the slope-intercept form [tex]\(y = mx + b\)[/tex].
Starting from the original equation:
[tex]\[
5x + 2y = 12
\][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[
2y = -5x + 12
\][/tex]
[tex]\[
y = -\frac{5}{2}x + 6
\][/tex]
Here, the slope [tex]\(m\)[/tex] is [tex]\(-\frac{5}{2}\)[/tex].
2. Determine the slope of the parallel line:
Since parallel lines have the same slope, the slope [tex]\(m\)[/tex] of the line we are looking for is also [tex]\(-\frac{5}{2}\)[/tex].
3. Use the point-slope form of the equation:
The point-slope form of a line's equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\(m = -\frac{5}{2}\)[/tex], [tex]\(x_1 = -2\)[/tex], and [tex]\(y_1 = 4\)[/tex].
Plug in the values:
[tex]\[
y - 4 = -\frac{5}{2}(x - (-2))
\][/tex]
Simplify the expression:
[tex]\[
y - 4 = -\frac{5}{2}(x + 2)
\][/tex]
Distribute [tex]\(-\frac{5}{2}\)[/tex]:
[tex]\[
y - 4 = -\frac{5}{2}x - 5
\][/tex]
Isolate [tex]\(y\)[/tex]:
[tex]\[
y = -\frac{5}{2}x - 5 + 4
\][/tex]
Simplify further:
[tex]\[
y = -\frac{5}{2}x - 1
\][/tex]
The correct equation of the line that is parallel to [tex]\(5x + 2y = 12\)[/tex] and passes through the point [tex]\((-2, 4)\)[/tex] is:
[tex]\[
y = -\frac{5}{2}x - 1
\][/tex]
Thus, among the provided options, the correct answer is:
[tex]\[
\boxed{y = -\frac{5}{2}x - 1}
\][/tex]
