To find the equation of a line that is parallel to a given line and passes through a specific point, we need to follow these steps:
1. Determine the slope of the given line: The equation of the given line is [tex]\(5x + 2y = 12\)[/tex]. First, we need to rewrite this equation in slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
[tex]\[
5x + 2y = 12
\][/tex]
Subtract [tex]\(5x\)[/tex] from both sides:
[tex]\[
2y = -5x + 12
\][/tex]
Divide everything by 2 to solve for [tex]\(y\)[/tex]:
[tex]\[
y = \left(-\frac{5}{2}\right)x + 6
\][/tex]
So, the slope ([tex]\(m\)[/tex]) of the given line is [tex]\(-\frac{5}{2}\)[/tex].
2. Use the slope of the parallel line: Since parallel lines have the same slope, the slope of the line we are looking for will also be [tex]\(-\frac{5}{2}\)[/tex].
3. Use the point-slope form of the equation of a line: The point-slope form is given by [tex]\[y - y_1 = m(x - x_1)\][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line. Here, the point is [tex]\((-2, 4)\)[/tex], and the slope [tex]\(m\)[/tex] is [tex]\(-\frac{5}{2}\)[/tex].
4. Substitute the known values into the point-slope form equation:
[tex]\[
y - 4 = \left(-\frac{5}{2}\right)(x - (-2))
\][/tex]
Simplify inside the parentheses:
[tex]\[
y - 4 = \left(-\frac{5}{2}\right)(x + 2)
\][/tex]
Distribute the slope [tex]\(-\frac{5}{2}\)[/tex]:
[tex]\[
y - 4 = \left(-\frac{5}{2}\right)x + \left(-\frac{5}{2}\right) \cdot 2
\][/tex]
Simplify the multiplication:
[tex]\[
y - 4 = -\frac{5}{2}x - 5
\][/tex]
Add 4 to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[
y = -\frac{5}{2}x - 5 + 4
\][/tex]
Simplify the constants:
[tex]\[
y = -\frac{5}{2}x - 1
\][/tex]
Therefore, the equation of the line that is parallel to [tex]\(5x + 2y = 12\)[/tex] and passes through the point [tex]\((-2,4)\)[/tex] is [tex]\(\boxed{y = -\frac{5}{2}x - 1}\)[/tex].
