Answer :
To find the equation of a line that is parallel to the given line [tex]\(2x + 5y = 10\)[/tex] and passes through the point [tex]\((-5,1)\)[/tex], follow these steps:
1. Identify the slope of the given line:
The given line [tex]\(2x + 5y = 10\)[/tex] is in standard form. To determine the slope of the line, convert it to slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
Rearrange the equation:
[tex]\[ 2x + 5y = 10 \][/tex]
[tex]\[ 5y = -2x + 10 \][/tex]
[tex]\[ y = -\frac{2}{5}x + 2 \][/tex]
So, the slope ([tex]\(m\)[/tex]) of the given line is [tex]\(-\frac{2}{5}\)[/tex].
2. Use the slope-point form of the equation of a line:
Since parallel lines have the same slope, the line we are looking for also has a slope of [tex]\(-\frac{2}{5}\)[/tex]. Now we use the point-slope form of the equation of a line, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\(m = -\frac{2}{5}\)[/tex] and the point is [tex]\((-5, 1)\)[/tex], so [tex]\(x_1 = -5\)[/tex] and [tex]\(y_1 = 1\)[/tex]. Substitute these into the point-slope form:
[tex]\[ y - 1 = -\frac{2}{5}(x + 5) \][/tex]
3. Identify the equivalent forms of the equation:
Next, simplify and compare this form to the given options.
One standard form is [tex]\(y - 1 = -\frac{2}{5}(x + 5)\)[/tex], which is directly listed as an option.
Convert [tex]\(y - 1 = -\frac{2}{5}(x + 5)\)[/tex] to slope-intercept form:
[tex]\[ y - 1 = -\frac{2}{5}x - 2 \][/tex]
[tex]\[ y = -\frac{2}{5}x - 1 \][/tex]
This gives us one option.
Now convert [tex]\(y = -\frac{2}{5}x - 1\)[/tex] to standard form:
[tex]\[ 5y = -2x - 5 \][/tex]
[tex]\[ 2x + 5y = -5 \][/tex]
This provides another equivalent form.
4. Compare with the given options:
- [tex]\(y = -\frac{2}{5}x - 1\)[/tex] is an accepted equation.
- [tex]\(2x + 5y = -5\)[/tex] is an accepted equation.
- [tex]\(y - 1 = -\frac{2}{5}(x + 5)\)[/tex] is an accepted equation.
The other options do not represent equivalent forms of the correct line's equation.
Thus, the correct equations are:
- [tex]\( y = -\frac{2}{5}x - 1 \)[/tex]
- [tex]\( 2x + 5y = -5 \)[/tex]
- [tex]\( y - 1 = -\frac{2}{5}(x + 5) \)[/tex]
1. Identify the slope of the given line:
The given line [tex]\(2x + 5y = 10\)[/tex] is in standard form. To determine the slope of the line, convert it to slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
Rearrange the equation:
[tex]\[ 2x + 5y = 10 \][/tex]
[tex]\[ 5y = -2x + 10 \][/tex]
[tex]\[ y = -\frac{2}{5}x + 2 \][/tex]
So, the slope ([tex]\(m\)[/tex]) of the given line is [tex]\(-\frac{2}{5}\)[/tex].
2. Use the slope-point form of the equation of a line:
Since parallel lines have the same slope, the line we are looking for also has a slope of [tex]\(-\frac{2}{5}\)[/tex]. Now we use the point-slope form of the equation of a line, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\(m = -\frac{2}{5}\)[/tex] and the point is [tex]\((-5, 1)\)[/tex], so [tex]\(x_1 = -5\)[/tex] and [tex]\(y_1 = 1\)[/tex]. Substitute these into the point-slope form:
[tex]\[ y - 1 = -\frac{2}{5}(x + 5) \][/tex]
3. Identify the equivalent forms of the equation:
Next, simplify and compare this form to the given options.
One standard form is [tex]\(y - 1 = -\frac{2}{5}(x + 5)\)[/tex], which is directly listed as an option.
Convert [tex]\(y - 1 = -\frac{2}{5}(x + 5)\)[/tex] to slope-intercept form:
[tex]\[ y - 1 = -\frac{2}{5}x - 2 \][/tex]
[tex]\[ y = -\frac{2}{5}x - 1 \][/tex]
This gives us one option.
Now convert [tex]\(y = -\frac{2}{5}x - 1\)[/tex] to standard form:
[tex]\[ 5y = -2x - 5 \][/tex]
[tex]\[ 2x + 5y = -5 \][/tex]
This provides another equivalent form.
4. Compare with the given options:
- [tex]\(y = -\frac{2}{5}x - 1\)[/tex] is an accepted equation.
- [tex]\(2x + 5y = -5\)[/tex] is an accepted equation.
- [tex]\(y - 1 = -\frac{2}{5}(x + 5)\)[/tex] is an accepted equation.
The other options do not represent equivalent forms of the correct line's equation.
Thus, the correct equations are:
- [tex]\( y = -\frac{2}{5}x - 1 \)[/tex]
- [tex]\( 2x + 5y = -5 \)[/tex]
- [tex]\( y - 1 = -\frac{2}{5}(x + 5) \)[/tex]
