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The point-slope form of the equation of the line that passes through [tex]\((-5,-1)\)[/tex] and [tex]\((10,-7)\)[/tex] is [tex]\( y + 7 = -\frac{2}{5}(x - 10) \)[/tex].

What is the standard form of the equation for this line?

A. [tex]\(2x - 5y = -15\)[/tex]

B. [tex]\(2x - 5y = -17\)[/tex]

C. [tex]\(2x + 5y = -15\)[/tex]

D. [tex]\(2x + 5y = -17\)[/tex]



Answer :

GinnyAnswer
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]

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