To convert the point-slope form of the line equation to the standard form, we need to follow a systematic sequence of algebraic steps. The given point-slope form of the equation is:
[tex]\[ y + 7 = -\frac{2}{5}(x - 10) \][/tex]
Let's break down the steps:
### Step-by-Step Solution
1. Initial form: The given point-slope form of the equation of the line is:
[tex]\[ y + 7 = -\frac{2}{5}(x - 10) \][/tex]
2. Clear the fraction by multiplying through by 5: This will help to eliminate the fraction and make the coefficients integers.
[tex]\[ 5(y + 7) = 5 \left( -\frac{2}{5} (x - 10) \right) \][/tex]
[tex]\[ 5(y + 7) = -2(x - 10) \][/tex]
3. Distribute and simplify:
[tex]\[ 5y + 35 = -2x + 20 \][/tex]
4. Rearrange terms to achieve the form [tex]\(ax + by = c\)[/tex]. We need to get all terms involving [tex]\(x\)[/tex] and [tex]\(y\)[/tex] on one side of the equation, and the constant on the other side.
[tex]\[ 2x + 5y = -15 \][/tex]
So, the standard form of the equation is:
[tex]\[ 2x + 5y = -15 \][/tex]
### Choosing the Correct Option
Among the provided choices:
- [tex]\( 2x - 5y = -15 \)[/tex]
- [tex]\( 2x - 5y = -17 \)[/tex]
- [tex]\( 2x + 5y = -15 \)[/tex]
- [tex]\( 2x + 5y = -17 \)[/tex]
The correct one is:
[tex]\[ 2x + 5y = -15 \][/tex]
Thus, the answer is:
[tex]\[ \boxed{2x + 5y = -15} \][/tex]
