To find the equation of the line passing through points [tex]\( C(3, -5) \)[/tex] and [tex]\( D(6, 0) \)[/tex] in standard form, we need to follow several steps.
1. Calculate the slope ([tex]\( m \)[/tex]) of the line:
The slope of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting [tex]\( C(3, -5) \)[/tex] and [tex]\( D(6, 0) \)[/tex] into the formula:
[tex]\[
m = \frac{0 - (-5)}{6 - 3} = \frac{5}{3}
\][/tex]
2. Using the point-slope form to find the equation of the line:
The point-slope form of a line's equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting [tex]\( (3, -5) \)[/tex] and [tex]\( m = \frac{5}{3} \)[/tex]:
[tex]\[
y - (-5) = \frac{5}{3}(x - 3)
\][/tex]
Simplifying this:
[tex]\[
y + 5 = \frac{5}{3}x - 5
\][/tex]
[tex]\[
y = \frac{5}{3}x - 10
\][/tex]
3. Convert the slope-intercept equation to standard form:
The standard form of a linear equation is [tex]\( Ax + By = C \)[/tex]. To convert [tex]\( y = \frac{5}{3}x - 10 \)[/tex] to this form, we need to eliminate the fractions by multiplying everything by 3:
[tex]\[
3y = 5x - 30
\][/tex]
Rearranging terms to get [tex]\( Ax + By = C \)[/tex]:
[tex]\[
5x - 3y = 30
\][/tex]
Therefore, the equation of the line [tex]\( CD \)[/tex] in standard form is [tex]\( 5x - 3y = 30 \)[/tex].
Checking the given options, we see:
- [tex]\( 5x + 3y = 18 \)[/tex]
- [tex]\( 5x - 3y = 30 \)[/tex]
- [tex]\( 5x - y = 30 \)[/tex]
- [tex]\( 5x + y = 18 \)[/tex]
The correct option is:
[tex]\[
\boxed{5x - 3y = 30}
\][/tex]
