Sure, let's work through the steps to find the equation of the line given the slope and a point on the line. We'll use this information to graph it as well.
### Given:
- A point on the line: (6, 2)
- Slope (m): -1
### Step-by-Step Solution:
1. Find the y-intercept (b):
The equation of a line in slope-intercept form is:
[tex]\[
y = mx + b
\][/tex]
We will use the given point (6, 2) to find the value of [tex]\( b \)[/tex]. Substitute the slope (m = -1) and the coordinates of the point (x = 6, y = 2) into the equation:
[tex]\[
2 = -1 \cdot 6 + b
\][/tex]
Simplify to find [tex]\( b \)[/tex]:
[tex]\[
2 = -6 + b
\][/tex]
Add 6 to both sides:
[tex]\[
2 + 6 = b
\][/tex]
[tex]\[
b = 8
\][/tex]
2. Write the equation of the line:
Now that we have the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex], we can write the equation of the line:
[tex]\[
y = -1x + 8
\][/tex]
or simply:
[tex]\[
y = -x + 8
\][/tex]
3. Graph the line:
To graph the line, we will identify two points. We already have one point (6, 2). We'll find another point to make the graphing easier.
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = -0 + 8 = 8
\][/tex]
So, the y-intercept is (0, 8).
- Choose another x value, say [tex]\( x = 4 \)[/tex]:
[tex]\[
y = -4 + 8 = 4
\][/tex]
So, another point is (4, 4).
Now, plot these points (0, 8), (4, 4), and (6, 2) on a coordinate plane and draw a straight line through them.
### Coordinates:
- Point 1: (0, 8) — Y-intercept.
- Point 2: (4, 4) — Another point on the line.
- Point 3: (6, 2) — Given point.
### Graph:
Use these points to draw the line accurately. The line will slope downward because the given slope is negative.
1. Start at point (0, 8) — This is the y-intercept.
2. Move down 1 unit and right 1 unit to (1, 7) using the slope of -1.
3. Repeat this slope movement for another point or directly connect the plotted points (0, 8), (4, 4), and (6, 2) with a straight line.
This is how you find the equation of a line given a point and a slope, and graph it accordingly.
