To find the equation of a line that is parallel to the line [tex]\(2x + 5y = 10\)[/tex] and passes through the point [tex]\((-5, 1)\)[/tex], we need to go through a step-by-step process. Let's begin:
1. Rewrite the given equation in slope-intercept form ([tex]\(y = mx + b\)[/tex]):
The given line is:
[tex]\[2x + 5y = 10\][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[5y = -2x + 10\][/tex]
[tex]\[y = -\frac{2}{5}x + 2\][/tex]
Here, the slope [tex]\(m\)[/tex] of the given line is [tex]\(-\frac{2}{5}\)[/tex].
2. Determine the slope of the parallel line:
Parallel lines have the same slope, so the slope [tex]\(m\)[/tex] of the new line will also be [tex]\(-\frac{2}{5}\)[/tex].
3. Use the point-slope form of a line equation:
The point-slope form is:
[tex]\[y - y_1 = m(x - x_1)\][/tex]
We are given the point [tex]\((-5, 1)\)[/tex], so [tex]\((x_1, y_1) = (-5, 1)\)[/tex] and [tex]\(m = -\frac{2}{5}\)[/tex].
Plugging these values into the point-slope form:
[tex]\[y - 1 = -\frac{2}{5}(x + 5)\][/tex]
4. Simplify the equation:
Distribute the slope [tex]\(-\frac{2}{5}\)[/tex] on the right side:
[tex]\[y - 1 = -\frac{2}{5}x -\frac{2}{5} \cdot 5\][/tex]
[tex]\[y - 1 = -\frac{2}{5}x - 2\][/tex]
Add 1 to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[y = -\frac{2}{5}x - 2 + 1\][/tex]
[tex]\[y = -\frac{2}{5}x - 1\][/tex]
5. Verify the equations presented in the options:
- [tex]\(y=-\frac{2}{5}x-1\)[/tex]
This matches our derived equation.
- [tex]\(2x + 5y = -5\)[/tex]
This does not match our derived equation. Verify by rewriting:
[tex]\[2x + 5y = -5\][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[5y = -2x - 5\][/tex]
[tex]\[y = -\frac{2}{5}x - 1\][/tex]
Notice that this fits the form [tex]\(y = -\frac{2}{5}x - 1\)[/tex].
- [tex]\(y=-\frac{2}{5} x-3\)[/tex]
This does not match our equation.
- [tex]\(2x + 5y = -15\)[/tex]
This does not match our equation. Verify by rewriting:
[tex]\[2x + 5y = -15\][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[5y = -2x - 15\][/tex]
[tex]\[y = -\frac{2}{5}x - 3\][/tex]
This does not match our derived equation.
- [tex]\(y-1=-\frac{2}{5}(x+5)\)[/tex]
This matches the point-slope form before we simplified it.
The correct equations are:
[tex]\[y = -\frac{2}{5}x - 1\][/tex]
and
[tex]\[y - 1 = -\frac{2}{5}(x + 5)\][/tex]
Thus, the correct options are:
[tex]\[y = -\frac{2}{5} x - 1\][/tex]
[tex]\[y - 1 = -\frac{2}{5}(x + 5)\][/tex]
