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Sagot :
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 follow these steps:
1. Identify the Slope of the Given Line:
The line equation [tex]\(2x + 5y = 10\)[/tex] can be rewritten in the slope-intercept form [tex]\(y = mx + b\)[/tex].
[tex]\[ 2x + 5y = 10 \][/tex]
[tex]\[ 5y = -2x + 10 \][/tex]
[tex]\[ y = -\frac{2}{5}x + 2 \][/tex]
From this, we see that the slope [tex]\(m\)[/tex] of the given line is [tex]\(-\frac{2}{5}\)[/tex].
2. Equation of Parallel Line:
Lines that are parallel have the same slope. Therefore, any line parallel to [tex]\(2x + 5y = 10\)[/tex] will have the slope [tex]\(-\frac{2}{5}\)[/tex].
3. Use the Point-Slope Form:
We use the point-slope form of the line equation, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Given the point [tex]\((-5, 1)\)[/tex] and slope [tex]\(-\frac{2}{5}\)[/tex], plug these into the formula:
[tex]\[ y - 1 = -\frac{2}{5}(x + 5) \][/tex]
Simplifying this equation:
[tex]\[ y - 1 = -\frac{2}{5}x - 2 \][/tex]
[tex]\[ y = -\frac{2}{5}x - 1 \][/tex]
Therefore, one equation of the line that is parallel to [tex]\(2x + 5y = 10\)[/tex] and passes through [tex]\((-5, 1)\)[/tex] is:
[tex]\[ y = -\frac{2}{5}x - 1 \][/tex]
4. Alternative Forms:
We can convert this into the general form of the equation. Let's check for the alternative forms that might match our requirements.
Another way to express the same line can be modifying [tex]\(2x + 5y = 10\)[/tex]:
[tex]\[ 2x + 5y = C \][/tex]
Substituting the point [tex]\((-5, 1)\)[/tex] into the line equation [tex]\(2x + 5y = C\)[/tex]:
[tex]\[ 2(-5) + 5(1) = C \][/tex]
[tex]\[ -10 + 5 = C \][/tex]
[tex]\[ C = -5 \][/tex]
Therefore, the equation of a line parallel to [tex]\(2x + 5y = 10\)[/tex] and passing through [tex]\((-5, 1)\)[/tex] is:
[tex]\[ 2x + 5y = -5 \][/tex]
5. Include Both Answer Formats:
There are other forms that represent the same equation but formatted differently.
Thus, permissible equations can be:
- [tex]\( y = -\frac{2}{5}x - 1 \)[/tex]
- [tex]\( 2x + 5y = -5 \)[/tex]
- [tex]\( y - 1 = -\frac{2}{5}(x + 5) \)[/tex]
6. Check Options:
- [tex]\( y = -\frac{2}{5} x - 1 \)[/tex]: This is one of the valid equations.
- [tex]\( 2 x + 5 y = -5 \)[/tex]: This is another valid equation.
- [tex]\( y = -\frac{2}{5} x - 3 \)[/tex]: This does not satisfy the conditions.
- [tex]\( 2 x + 5 y = -15 \)[/tex]: This does not satisfy the conditions.
- [tex]\( y - 1 = -\frac{2}{5} (x + 5) \)[/tex]: This is also a valid equation in a different form.
So, the correct equations that are parallel to [tex]\(2x + 5y = 10\)[/tex] and pass through the point [tex]\((-5, 1)\)[/tex] are:
1. [tex]\( y = -\frac{2}{5} x - 1 \)[/tex]
2. [tex]\( 2 x + 5 y = -5 \)[/tex]
3. [tex]\( y - 1 = -\frac{2}{5} (x + 5) \)[/tex]
1. Identify the Slope of the Given Line:
The line equation [tex]\(2x + 5y = 10\)[/tex] can be rewritten in the slope-intercept form [tex]\(y = mx + b\)[/tex].
[tex]\[ 2x + 5y = 10 \][/tex]
[tex]\[ 5y = -2x + 10 \][/tex]
[tex]\[ y = -\frac{2}{5}x + 2 \][/tex]
From this, we see that the slope [tex]\(m\)[/tex] of the given line is [tex]\(-\frac{2}{5}\)[/tex].
2. Equation of Parallel Line:
Lines that are parallel have the same slope. Therefore, any line parallel to [tex]\(2x + 5y = 10\)[/tex] will have the slope [tex]\(-\frac{2}{5}\)[/tex].
3. Use the Point-Slope Form:
We use the point-slope form of the line equation, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Given the point [tex]\((-5, 1)\)[/tex] and slope [tex]\(-\frac{2}{5}\)[/tex], plug these into the formula:
[tex]\[ y - 1 = -\frac{2}{5}(x + 5) \][/tex]
Simplifying this equation:
[tex]\[ y - 1 = -\frac{2}{5}x - 2 \][/tex]
[tex]\[ y = -\frac{2}{5}x - 1 \][/tex]
Therefore, one equation of the line that is parallel to [tex]\(2x + 5y = 10\)[/tex] and passes through [tex]\((-5, 1)\)[/tex] is:
[tex]\[ y = -\frac{2}{5}x - 1 \][/tex]
4. Alternative Forms:
We can convert this into the general form of the equation. Let's check for the alternative forms that might match our requirements.
Another way to express the same line can be modifying [tex]\(2x + 5y = 10\)[/tex]:
[tex]\[ 2x + 5y = C \][/tex]
Substituting the point [tex]\((-5, 1)\)[/tex] into the line equation [tex]\(2x + 5y = C\)[/tex]:
[tex]\[ 2(-5) + 5(1) = C \][/tex]
[tex]\[ -10 + 5 = C \][/tex]
[tex]\[ C = -5 \][/tex]
Therefore, the equation of a line parallel to [tex]\(2x + 5y = 10\)[/tex] and passing through [tex]\((-5, 1)\)[/tex] is:
[tex]\[ 2x + 5y = -5 \][/tex]
5. Include Both Answer Formats:
There are other forms that represent the same equation but formatted differently.
Thus, permissible equations can be:
- [tex]\( y = -\frac{2}{5}x - 1 \)[/tex]
- [tex]\( 2x + 5y = -5 \)[/tex]
- [tex]\( y - 1 = -\frac{2}{5}(x + 5) \)[/tex]
6. Check Options:
- [tex]\( y = -\frac{2}{5} x - 1 \)[/tex]: This is one of the valid equations.
- [tex]\( 2 x + 5 y = -5 \)[/tex]: This is another valid equation.
- [tex]\( y = -\frac{2}{5} x - 3 \)[/tex]: This does not satisfy the conditions.
- [tex]\( 2 x + 5 y = -15 \)[/tex]: This does not satisfy the conditions.
- [tex]\( y - 1 = -\frac{2}{5} (x + 5) \)[/tex]: This is also a valid equation in a different form.
So, the correct equations that are parallel to [tex]\(2x + 5y = 10\)[/tex] and pass through the point [tex]\((-5, 1)\)[/tex] are:
1. [tex]\( y = -\frac{2}{5} x - 1 \)[/tex]
2. [tex]\( 2 x + 5 y = -5 \)[/tex]
3. [tex]\( y - 1 = -\frac{2}{5} (x + 5) \)[/tex]
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