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Sagot :
To graph a line that contains the point [tex]\((-7, -4)\)[/tex] and has a slope of [tex]\(-\frac{2}{3}\)[/tex], we will follow these steps:
### Step 1: Identify the point and the slope
- The point given is [tex]\((-7,-4)\)[/tex].
- The slope [tex]\(m = -\frac{2}{3}\)[/tex].
### Step 2: Use the point-slope form of a line equation
The point-slope form of a line equation is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\( (x_1, y_1) \)[/tex] is the point [tex]\((-7, -4)\)[/tex] and [tex]\( m = -\frac{2}{3} \)[/tex].
Substitute these values into the equation:
[tex]\[ y - (-4) = -\frac{2}{3}(x - (-7)) \][/tex]
Simplify the equation:
[tex]\[ y + 4 = -\frac{2}{3}(x + 7) \][/tex]
### Step 3: Convert to slope-intercept form
The slope-intercept form of a line equation is [tex]\( y = mx + b \)[/tex].
Distribute the slope [tex]\( -\frac{2}{3} \)[/tex]:
[tex]\[ y + 4 = -\frac{2}{3}x - \frac{2}{3} \cdot 7 \][/tex]
[tex]\[ y + 4 = -\frac{2}{3}x - \frac{14}{3} \][/tex]
To isolate [tex]\( y \)[/tex], subtract 4 from both sides of the equation:
[tex]\[ y = -\frac{2}{3}x - \frac{14}{3} - 4 \][/tex]
Convert 4 to a fraction with a common denominator (denominator 3):
[tex]\[ 4 = \frac{12}{3} \][/tex]
[tex]\[ y = -\frac{2}{3}x - \frac{14}{3} - \frac{12}{3} \][/tex]
[tex]\[ y = -\frac{2}{3}x - \frac{26}{3} \][/tex]
Thus, the equation of the line in slope-intercept form is:
[tex]\[ y = -\frac{2}{3}x - \frac{26}{3} \][/tex]
### Step 4: Identify the slope and y-intercept from the equation
From the slope-intercept form equation [tex]\( y = mx + b \)[/tex]:
- The slope [tex]\( m = -\frac{2}{3} \)[/tex].
- The y-intercept [tex]\( b = -\frac{26}{3} \)[/tex].
### Step 5: Plot the point and draw the line
1. Start by plotting the point [tex]\((-7, -4)\)[/tex] on the coordinate plane.
2. To find another point, use the slope [tex]\( -\frac{2}{3} \)[/tex]. This means for every 3 units you move to the right, you move 2 units down (due to the negative sign).
However, note the slope and y-intercept as calculated can also directly provide additional points or assist in plotting the line more accurately:
- The slope: [tex]\( -0.666... \)[/tex] (approximately)
- The y-intercept: [tex]\( -8.666... \)[/tex] (approximately)
### Summary:
You successfully graphed the line by determining the equation [tex]\( y = -\frac{2}{3}x - \frac{26}{3} \)[/tex], knowing the slope, identifying the y-intercept, and finally using this information to correctly draw the graph starting from point [tex]\((-7, -4)\)[/tex]. Remember to plot ([tex]\(0, -8.666...)), then connect these points with a straight line illustrating the slope \(-\frac{2}{3}\)[/tex].
### Step 1: Identify the point and the slope
- The point given is [tex]\((-7,-4)\)[/tex].
- The slope [tex]\(m = -\frac{2}{3}\)[/tex].
### Step 2: Use the point-slope form of a line equation
The point-slope form of a line equation is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\( (x_1, y_1) \)[/tex] is the point [tex]\((-7, -4)\)[/tex] and [tex]\( m = -\frac{2}{3} \)[/tex].
Substitute these values into the equation:
[tex]\[ y - (-4) = -\frac{2}{3}(x - (-7)) \][/tex]
Simplify the equation:
[tex]\[ y + 4 = -\frac{2}{3}(x + 7) \][/tex]
### Step 3: Convert to slope-intercept form
The slope-intercept form of a line equation is [tex]\( y = mx + b \)[/tex].
Distribute the slope [tex]\( -\frac{2}{3} \)[/tex]:
[tex]\[ y + 4 = -\frac{2}{3}x - \frac{2}{3} \cdot 7 \][/tex]
[tex]\[ y + 4 = -\frac{2}{3}x - \frac{14}{3} \][/tex]
To isolate [tex]\( y \)[/tex], subtract 4 from both sides of the equation:
[tex]\[ y = -\frac{2}{3}x - \frac{14}{3} - 4 \][/tex]
Convert 4 to a fraction with a common denominator (denominator 3):
[tex]\[ 4 = \frac{12}{3} \][/tex]
[tex]\[ y = -\frac{2}{3}x - \frac{14}{3} - \frac{12}{3} \][/tex]
[tex]\[ y = -\frac{2}{3}x - \frac{26}{3} \][/tex]
Thus, the equation of the line in slope-intercept form is:
[tex]\[ y = -\frac{2}{3}x - \frac{26}{3} \][/tex]
### Step 4: Identify the slope and y-intercept from the equation
From the slope-intercept form equation [tex]\( y = mx + b \)[/tex]:
- The slope [tex]\( m = -\frac{2}{3} \)[/tex].
- The y-intercept [tex]\( b = -\frac{26}{3} \)[/tex].
### Step 5: Plot the point and draw the line
1. Start by plotting the point [tex]\((-7, -4)\)[/tex] on the coordinate plane.
2. To find another point, use the slope [tex]\( -\frac{2}{3} \)[/tex]. This means for every 3 units you move to the right, you move 2 units down (due to the negative sign).
However, note the slope and y-intercept as calculated can also directly provide additional points or assist in plotting the line more accurately:
- The slope: [tex]\( -0.666... \)[/tex] (approximately)
- The y-intercept: [tex]\( -8.666... \)[/tex] (approximately)
### Summary:
You successfully graphed the line by determining the equation [tex]\( y = -\frac{2}{3}x - \frac{26}{3} \)[/tex], knowing the slope, identifying the y-intercept, and finally using this information to correctly draw the graph starting from point [tex]\((-7, -4)\)[/tex]. Remember to plot ([tex]\(0, -8.666...)), then connect these points with a straight line illustrating the slope \(-\frac{2}{3}\)[/tex].
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