To determine the equation of the line that is parallel to the given line [tex]\(5x + 2y = 12\)[/tex] and passes through the point [tex]\((-2, 4)\)[/tex], we can follow these steps:
1. Determine the slope of the given line:
To find the slope, we first convert the given line equation into the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope.
Given equation:
[tex]\[ 5x + 2y = 12 \][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[ 2y = -5x + 12 \][/tex]
[tex]\[ y = -\frac{5}{2}x + 6 \][/tex]
From this form, we can see that the slope ([tex]\(m\)[/tex]) of the given line is [tex]\(-\frac{5}{2}\)[/tex].
2. Identify the slope of the parallel line:
A line that is parallel to the given line will have the same slope. Therefore, the parallel line will also have the slope [tex]\(m = -\frac{5}{2}\)[/tex].
3. Use the point-slope form to write the equation of the parallel line:
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] and the point [tex]\((-2, 4)\)[/tex] will be [tex]\((x_1, y_1)\)[/tex].
Substituting these values into the point-slope form:
[tex]\[ y - 4 = -\frac{5}{2}(x + 2) \][/tex]
Simplify this equation step by step:
[tex]\[ y - 4 = -\frac{5}{2}x - 5 \][/tex]
[tex]\[ y = -\frac{5}{2}x - 5 + 4 \][/tex]
[tex]\[ y = -\frac{5}{2}x - 1 \][/tex]
4. Verify the correct option:
Comparing the obtained equation with the options provided:
- [tex]\( y = -\frac{5}{2}x - 1 \)[/tex] (this matches our derived equation)
- [tex]\( y = -\frac{5}{2}x + 5 \)[/tex]
- [tex]\( y = \frac{2}{5}x - 1 \)[/tex]
- [tex]\( y = \frac{2}{5}x + 5 \)[/tex]
The correct equation of the line that is parallel to the line [tex]\(5x + 2y = 12\)[/tex] and passes through the point [tex]\((-2, 4)\)[/tex] is:
[tex]\[ y = -\frac{5}{2}x - 1 \][/tex]
Thus, the correct answer is the first option:
[tex]\[ \boxed{y = -\frac{5}{2} x - 1} \][/tex]
