To graph the line passing through the point (-6, 1) with a slope of [tex]\( m = -6 \)[/tex], follow these steps:
1. Identify the slope and the initial point:
- Slope ([tex]\( m \)[/tex]): -6
- Point ([tex]\( x_1, y_1 \)[/tex]): (-6, 1)
2. Find another point on the line:
We'll use the point-slope form of the line equation:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substitute the given point (-6, 1) and slope (-6) into the equation. To find another point, choose an arbitrary x-value different from -6; here, we choose [tex]\( x_2 = -5 \)[/tex].
3. Calculate the corresponding y-value for the chosen x-value [tex]\( x_2 = -5 \)[/tex]:
[tex]\[
y - 1 = -6(x - (-6))
\][/tex]
Substitute [tex]\( x = -5 \)[/tex]:
[tex]\[
y - 1 = -6(-5 + 6)
\][/tex]
Simplify inside the parentheses:
[tex]\[
y - 1 = -6(1)
\][/tex]
[tex]\[
y - 1 = -6
\][/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[
y = -6 + 1
\][/tex]
[tex]\[
y = -5
\][/tex]
So, the point [tex]\( (x_2, y_2) \)[/tex] is [tex]\((-5, -5)\)[/tex].
4. Find the y-intercept (b):
Recall the slope-intercept form [tex]\( y = mx + b \)[/tex]. We use the initial point (-6, 1) to find [tex]\( b \)[/tex]:
[tex]\[
y = mx + b
\][/tex]
Substitute [tex]\( y = 1 \)[/tex], [tex]\( m = -6 \)[/tex], and [tex]\( x = -6 \)[/tex]:
[tex]\[
1 = -6(-6) + b
\][/tex]
Simplify:
[tex]\[
1 = 36 + b
\][/tex]
Solve for [tex]\( b \)[/tex]:
[tex]\[
b = 1 - 36
\][/tex]
[tex]\[
b = -35
\][/tex]
Thus, the equation of the line is:
[tex]\[
y = -6x - 35
\][/tex]
5. Summarize the key points:
- Slope ([tex]\( m \)[/tex]): -6
- Given Point: (-6, 1)
- Calculated Point: (-5, -5)
- Y-intercept ([tex]\( b \)[/tex]): -35
Using this information, you can graph the line by:
- Plotting the given point (-6, 1)
- Plotting the calculated point (-5, -5)
- Drawing a straight line through these points
- Ensuring the line extends in both directions and crosses the y-axis at -35.
These detailed steps and calculations ensure that the line is accurately graphed according to the given conditions.
