To find the equation of a line parallel to [tex]\( 2x + 5y = 10 \)[/tex] that passes through the point [tex]\((-5, 1)\)[/tex], let's proceed through the steps methodically.
### Step 1: Determine the slope of the given line
First, we need to rewrite the given line equation [tex]\( 2x + 5y = 10 \)[/tex] in slope-intercept form [tex]\( y = mx + b \)[/tex]. This will help us identify the slope [tex]\( m \)[/tex].
[tex]\[ 2x + 5y = 10 \][/tex]
[tex]\[ 5y = -2x + 10 \][/tex]
[tex]\[ y = -\frac{2}{5}x + 2 \][/tex]
From the equation [tex]\( y = -\frac{2}{5}x + 2 \)[/tex], it is clear that the slope [tex]\( m \)[/tex] is [tex]\( -\frac{2}{5} \)[/tex]. Any line parallel to this line must have the same slope.
### Step 2: Use the point-slope form
We know the parallel line must have the same slope [tex]\( m = -\frac{2}{5} \)[/tex] and it passes through the point [tex]\((-5, 1)\)[/tex]. We use the point-slope form of a line equation, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] is the point [tex]\((-5, 1)\)[/tex], and [tex]\( m = -\frac{2}{5} \)[/tex].
Substitute these values into the point-slope form:
[tex]\[ y - 1 = -\frac{2}{5}(x - (-5)) \][/tex]
[tex]\[ y - 1 = -\frac{2}{5}(x + 5) \][/tex]
### Step 3: Simplify and convert to slope-intercept form (if necessary)
To verify any alternative forms like slope-intercept form, we can simplify further:
[tex]\[ y - 1 = -\frac{2}{5}x - \frac{2}{5} \cdot 5 \][/tex]
[tex]\[ y - 1 = -\frac{2}{5}x - 2 \][/tex]
[tex]\[ y = -\frac{2}{5}x - 1 \][/tex]
### Verify the Answer Choices
Now let's check the given options to see which ones match the derived equations:
1. [tex]\( y = -\frac{2}{5}x - 1 \)[/tex]
- This directly matches the slope-intercept form we derived. Correct.
2. [tex]\( 2x + 5y = -5 \)[/tex]
- Rewriting in slope-intercept form:
[tex]\[ 5y = -2x - 5 \][/tex]
[tex]\[ y = -\frac{2}{5}x - 1 \][/tex]
- This does not match our slope and y-intercept. Incorrect.
3. [tex]\( y = -\frac{2}{5}x - 3 \)[/tex]
- This has the correct slope, but the y-intercept is incorrect. Incorrect.
4. [tex]\( 24 - 5y = -15 \)[/tex]
- Rewriting in slope-intercept form:
[tex]\[ 24 + 15 = 5y \][/tex]
[tex]\[ 5y = 39 \][/tex]
[tex]\[ y = \frac{39}{5} \][/tex]
- This does not match our slope. Incorrect.
5. [tex]\( y - 1 = -\frac{2}{5}(x + 5) \)[/tex]
- This matches the original point-slope form we derived. Correct.
Therefore, the correct answers are:
- [tex]\( y = -\frac{2}{5}x - 1 \)[/tex]
- [tex]\( y - 1 = -\frac{2}{5}(x + 5) \)[/tex]
