To determine the equation of the line passing through the points [tex]\( J(-3, 11) \)[/tex] and [tex]\( K(1, -3) \)[/tex] in standard form, we will go through the following steps:
### Step 1: Calculate the Slope 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 given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substitute the coordinates of points [tex]\( J \)[/tex] and [tex]\( K \)[/tex]:
[tex]\[
m = \frac{-3 - 11}{1 - (-3)} = \frac{-14}{4} = -3.5
\][/tex]
So, the slope of the line is [tex]\( m = -3.5 \)[/tex].
### Step 2: Calculate the Y-Intercept
The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. We can use the coordinates of point [tex]\( J(-3, 11) \)[/tex] to find [tex]\( b \)[/tex]:
[tex]\[
11 = -3.5(-3) + b
\][/tex]
[tex]\[
11 = 10.5 + b
\][/tex]
[tex]\[
b = 11 - 10.5 = 0.5
\][/tex]
So, the y-intercept is [tex]\( b = 0.5 \)[/tex], and the equation of the line in slope-intercept form is:
[tex]\[
y = -3.5x + 0.5
\][/tex]
### Step 3: Convert to Standard Form
The standard form of a line is [tex]\( Ax + By = C \)[/tex]. To convert from the slope-intercept form, we rearrange the equation:
[tex]\[
y = -3.5x + 0.5
\][/tex]
[tex]\[
3.5x + y = 0.5
\][/tex]
To eliminate the decimal, we multiply every term by 2:
[tex]\[
2(3.5x) + 2(y) = 2(0.5)
\][/tex]
[tex]\[
7x + 2y = 1
\][/tex]
### Step 4: Compare with Given Choices
Now we compare this standard form equation with the given choices:
1. [tex]\( 7x + 2y = -1 \)[/tex]
2. [tex]\( 7x + 2y = 1 \)[/tex]
3. [tex]\( 14x + 4y = -1 \)[/tex]
4. [tex]\( 14x + 4y = 1 \)[/tex]
We see that the equation [tex]\( 7x + 2y = 1 \)[/tex] matches our derived equation.
### Conclusion
The equation of line [tex]\( J K \)[/tex] in standard form is:
[tex]\[
\boxed{7x + 2y = 1}
\][/tex]
