Answer :
To graph a line that contains the point [tex]\((3, -6)\)[/tex] and has a slope of [tex]\(-\frac{1}{2}\)[/tex], let's go through the steps systematically.
### Step 1: Understand the Line Equation
The general form of the equation of a line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
### Step 2: Substitute the Slope
In our case, the slope ([tex]\( m \)[/tex]) is [tex]\(-\frac{1}{2}\)[/tex]. So the equation becomes:
[tex]\[ y = -\frac{1}{2}x + b \][/tex]
### Step 3: Use the Given Point to Find the Y-intercept
We know the line passes through the point [tex]\((3, -6)\)[/tex]. We substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -6\)[/tex] into the equation to solve for [tex]\( b \)[/tex]:
[tex]\[ -6 = -\frac{1}{2}(3) + b \][/tex]
Solve for [tex]\( b \)[/tex]:
[tex]\[ -6 = -\frac{3}{2} + b \][/tex]
Add [tex]\(\frac{3}{2}\)[/tex] to both sides to isolate [tex]\( b \)[/tex]:
[tex]\[ -6 + \frac{3}{2} = b \][/tex]
Convert [tex]\(-6\)[/tex] to a fraction with the same denominator for easier addition:
[tex]\[ -\frac{12}{2} + \frac{3}{2} = b \][/tex]
Combine the fractions:
[tex]\[ b = -\frac{9}{2} \][/tex]
### Step 4: Write the Full Equation of the Line
Substitute [tex]\( b = -\frac{9}{2} \)[/tex] back into the equation:
[tex]\[ y = -\frac{1}{2}x - \frac{9}{2} \][/tex]
### Step 5: Plot the Line
1. Start by plotting the point [tex]\((3, -6)\)[/tex].
2. Use the slope to find another point. Since the slope is [tex]\(-\frac{1}{2}\)[/tex], it means that for each unit increase in [tex]\(x\)[/tex], [tex]\(y\)[/tex] decreases by [tex]\(\frac{1}{2}\)[/tex].
Another way to use the slope is:
- Move 2 units to the right (positive direction along [tex]\(x\)[/tex]-axis).
- Move 1 unit down (negative direction along [tex]\(y\)[/tex]-axis).
So starting from [tex]\((3, -6)\)[/tex]:
- Move 2 units to the right: [tex]\(3 + 2 = 5\)[/tex]
- Move 1 unit down: [tex]\(-6 - 1 = -7\)[/tex]
Thus, another point on the line is [tex]\((5, -7)\)[/tex].
3. Plot both points [tex]\((3, -6)\)[/tex] and [tex]\((5, -7)\)[/tex] on the graph.
4. Draw a straight line through these points, extending it in both directions.
### Step 6: Label and Finalize the Graph
- Label the axes [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
- Indicate the line equation [tex]\( y = -\frac{1}{2}x - \frac{9}{2} \)[/tex].
- Include grid lines and any other necessary markings.
By following these directions, you’ll have a graph of the line that contains the point [tex]\((3, -6)\)[/tex] with a slope of [tex]\(-\frac{1}{2}\)[/tex].
### Step 1: Understand the Line Equation
The general form of the equation of a line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
### Step 2: Substitute the Slope
In our case, the slope ([tex]\( m \)[/tex]) is [tex]\(-\frac{1}{2}\)[/tex]. So the equation becomes:
[tex]\[ y = -\frac{1}{2}x + b \][/tex]
### Step 3: Use the Given Point to Find the Y-intercept
We know the line passes through the point [tex]\((3, -6)\)[/tex]. We substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -6\)[/tex] into the equation to solve for [tex]\( b \)[/tex]:
[tex]\[ -6 = -\frac{1}{2}(3) + b \][/tex]
Solve for [tex]\( b \)[/tex]:
[tex]\[ -6 = -\frac{3}{2} + b \][/tex]
Add [tex]\(\frac{3}{2}\)[/tex] to both sides to isolate [tex]\( b \)[/tex]:
[tex]\[ -6 + \frac{3}{2} = b \][/tex]
Convert [tex]\(-6\)[/tex] to a fraction with the same denominator for easier addition:
[tex]\[ -\frac{12}{2} + \frac{3}{2} = b \][/tex]
Combine the fractions:
[tex]\[ b = -\frac{9}{2} \][/tex]
### Step 4: Write the Full Equation of the Line
Substitute [tex]\( b = -\frac{9}{2} \)[/tex] back into the equation:
[tex]\[ y = -\frac{1}{2}x - \frac{9}{2} \][/tex]
### Step 5: Plot the Line
1. Start by plotting the point [tex]\((3, -6)\)[/tex].
2. Use the slope to find another point. Since the slope is [tex]\(-\frac{1}{2}\)[/tex], it means that for each unit increase in [tex]\(x\)[/tex], [tex]\(y\)[/tex] decreases by [tex]\(\frac{1}{2}\)[/tex].
Another way to use the slope is:
- Move 2 units to the right (positive direction along [tex]\(x\)[/tex]-axis).
- Move 1 unit down (negative direction along [tex]\(y\)[/tex]-axis).
So starting from [tex]\((3, -6)\)[/tex]:
- Move 2 units to the right: [tex]\(3 + 2 = 5\)[/tex]
- Move 1 unit down: [tex]\(-6 - 1 = -7\)[/tex]
Thus, another point on the line is [tex]\((5, -7)\)[/tex].
3. Plot both points [tex]\((3, -6)\)[/tex] and [tex]\((5, -7)\)[/tex] on the graph.
4. Draw a straight line through these points, extending it in both directions.
### Step 6: Label and Finalize the Graph
- Label the axes [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
- Indicate the line equation [tex]\( y = -\frac{1}{2}x - \frac{9}{2} \)[/tex].
- Include grid lines and any other necessary markings.
By following these directions, you’ll have a graph of the line that contains the point [tex]\((3, -6)\)[/tex] with a slope of [tex]\(-\frac{1}{2}\)[/tex].
