To find the equation of the line passing through points [tex]\( J(-3, 11) \)[/tex] and [tex]\( K(1, -3) \)[/tex] in standard form, we need to go through several steps:
1. Calculate the slope (m) of the line:
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the coordinates of points [tex]\( J \)[/tex] and [tex]\( K \)[/tex]:
[tex]\[
m = \frac{-3 - 11}{1 - (-3)}
\][/tex]
Simplify the subtraction and addition:
[tex]\[
m = \frac{-14}{4} = -3.5
\][/tex]
2. Calculate the y-intercept (b):
We use the point-slope form of the line equation [tex]\( y = mx + b \)[/tex]. Choose point [tex]\( J \)[/tex] for the calculation:
[tex]\[
11 = (-3.5)(-3) + b
\][/tex]
Simplify and solve for [tex]\( b \)[/tex]:
[tex]\[
11 = 10.5 + b \\
b = 11 - 10.5 \\
b = 0.5
\][/tex]
So the equation of the line in slope-intercept form is:
[tex]\[
y = -3.5x + 0.5
\][/tex]
3. Convert the equation to standard form [tex]\( Ax + By = C \)[/tex]:
Start with:
[tex]\[
y = -3.5x + 0.5
\][/tex]
To eliminate the decimal in the slope, multiply every term by 2:
[tex]\[
2y = -7x + 1
\][/tex]
Rearrange to standard form:
[tex]\[
7x + 2y = 1
\][/tex]
4. Compare the simplified standard form with the given options:
The given problem options are:
- [tex]\( 7x + 2y = -1 \)[/tex]
- [tex]\( 7x + 2y = 1 \)[/tex]
- [tex]\( 14x + 4y = -1 \)[/tex]
- [tex]\( 14x + 4y = 1 \)[/tex]
We can see that the correct standard form from our derivation is:
[tex]\[
7x + 2y = 1
\][/tex]
Therefore, the correct equation of line [tex]\( JK \)[/tex] in standard form is:
[tex]\[
7x + 2y = 1
\][/tex]
