Answer :
To write the equation of a line in standard form, we start by using the given points [tex]\((8, -1)\)[/tex] and [tex]\((2, -5)\)[/tex]. The point-slope form of the line, given as [tex]\(y + 1 = \frac{2}{3}(x - 8)\)[/tex], will help us transition to the standard form [tex]\(Ax + By = C\)[/tex].
### Step-by-Step Solution:
1. Start with the point-slope form:
[tex]\[ y + 1 = \frac{2}{3}(x - 8) \][/tex]
2. Eliminate the fraction by multiplying all terms by 3:
[tex]\[ 3(y + 1) = 2(x - 8) \][/tex]
3. Distribute both sides:
[tex]\[ 3y + 3 = 2x - 16 \][/tex]
4. Get all terms on one side to form [tex]\(Ax + By = C\)[/tex]:
[tex]\[ 3y + 3 - 2x = -16 \][/tex]
[tex]\[ -2x + 3y = -16 - 3 \][/tex]
[tex]\[ -2x + 3y = -19 \][/tex]
5. Multiply by [tex]\(-1\)[/tex] to make the coefficient of [tex]\(x\)[/tex] positive:
[tex]\[ 2x - 3y = 19 \][/tex]
Thus, the equation of the line in standard form is:
[tex]\[ \boxed{2}x + \boxed{(-3)}y = \boxed{19} \][/tex]
So, the blanks filled would be:
2 [tex]\(x\)[/tex] + (-3) [tex]\(y\)[/tex] = 19
### Step-by-Step Solution:
1. Start with the point-slope form:
[tex]\[ y + 1 = \frac{2}{3}(x - 8) \][/tex]
2. Eliminate the fraction by multiplying all terms by 3:
[tex]\[ 3(y + 1) = 2(x - 8) \][/tex]
3. Distribute both sides:
[tex]\[ 3y + 3 = 2x - 16 \][/tex]
4. Get all terms on one side to form [tex]\(Ax + By = C\)[/tex]:
[tex]\[ 3y + 3 - 2x = -16 \][/tex]
[tex]\[ -2x + 3y = -16 - 3 \][/tex]
[tex]\[ -2x + 3y = -19 \][/tex]
5. Multiply by [tex]\(-1\)[/tex] to make the coefficient of [tex]\(x\)[/tex] positive:
[tex]\[ 2x - 3y = 19 \][/tex]
Thus, the equation of the line in standard form is:
[tex]\[ \boxed{2}x + \boxed{(-3)}y = \boxed{19} \][/tex]
So, the blanks filled would be:
2 [tex]\(x\)[/tex] + (-3) [tex]\(y\)[/tex] = 19
