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Sagot :
To determine which equation represents the line passing through the points [tex]\((-8, 11)\)[/tex] and [tex]\(\left(4, \frac{7}{2}\right)\)[/tex], let's go through the necessary steps to find the equation of the line in slope-intercept form, [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
### Step 1: Calculate the Slope ([tex]\(m\)[/tex])
The slope [tex]\(m\)[/tex] of a line passing through 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]
For the points [tex]\((-8, 11)\)[/tex] and [tex]\(\left(4, \frac{7}{2}\right)\)[/tex]:
- [tex]\(x_1 = -8\)[/tex]
- [tex]\(y_1 = 11\)[/tex]
- [tex]\(x_2 = 4\)[/tex]
- [tex]\(y_2 = \frac{7}{2}\)[/tex]
Plugging in these values, the slope is:
[tex]\[ m = \frac{\frac{7}{2} - 11}{4 - (-8)} \][/tex]
[tex]\[ m = \frac{\frac{7}{2} - \frac{22}{2}}{4 + 8} \][/tex]
[tex]\[ m = \frac{\frac{7 - 22}{2}}{12} \][/tex]
[tex]\[ m = \frac{\frac{-15}{2}}{12} \][/tex]
[tex]\[ m = \frac{-15}{2} \times \frac{1}{12} \][/tex]
[tex]\[ m = -\frac{15}{24} \][/tex]
[tex]\[ m = -\frac{5}{8} \][/tex]
So, the slope [tex]\(m\)[/tex] of the line is [tex]\(-\frac{5}{8}\)[/tex].
### Step 2: Calculate the Y-Intercept ([tex]\(b\)[/tex])
Using the slope-intercept form [tex]\(y = mx + b\)[/tex], we substitute one of the points and the slope to solve for [tex]\(b\)[/tex].
Using the point [tex]\((-8, 11)\)[/tex] and the slope [tex]\(m = -\frac{5}{8}\)[/tex]:
[tex]\[ y = mx + b \][/tex]
[tex]\[ 11 = -\frac{5}{8}(-8) + b \][/tex]
[tex]\[ 11 = \frac{40}{8} + b \][/tex]
[tex]\[ 11 = 5 + b \][/tex]
[tex]\[ b = 11 - 5 \][/tex]
[tex]\[ b = 6 \][/tex]
So, the y-intercept [tex]\(b\)[/tex] is 6.
### Step 3: Form the Equation of the Line
Now that we have the slope and the y-intercept, we can write the equation of the line:
[tex]\[ y = -\frac{5}{8}x + 6 \][/tex]
### Step 4: Match the Equation with the Given Choices
We need to check which of the provided equations match our derived equation:
1. [tex]\( y = -\frac{5}{8}x + 6 \)[/tex]
2. [tex]\( y = -\frac{5}{8}x + 16 \)[/tex]
3. [tex]\( y = -\frac{15}{2}x - 49 \)[/tex]
4. [tex]\( y = -\frac{15}{2}x + 71 \)[/tex]
From the calculations, it is clear that the equation:
[tex]\[ y = -\frac{5}{8}x + 6 \][/tex]
matches the equation we derived.
Thus, the correct equation representing the line passing through [tex]\((-8, 11)\)[/tex] and [tex]\(\left(4, \frac{7}{2}\right)\)[/tex] is:
[tex]\[ y = -\frac{5}{8}x + 6 \][/tex]
### Step 1: Calculate the Slope ([tex]\(m\)[/tex])
The slope [tex]\(m\)[/tex] of a line passing through 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]
For the points [tex]\((-8, 11)\)[/tex] and [tex]\(\left(4, \frac{7}{2}\right)\)[/tex]:
- [tex]\(x_1 = -8\)[/tex]
- [tex]\(y_1 = 11\)[/tex]
- [tex]\(x_2 = 4\)[/tex]
- [tex]\(y_2 = \frac{7}{2}\)[/tex]
Plugging in these values, the slope is:
[tex]\[ m = \frac{\frac{7}{2} - 11}{4 - (-8)} \][/tex]
[tex]\[ m = \frac{\frac{7}{2} - \frac{22}{2}}{4 + 8} \][/tex]
[tex]\[ m = \frac{\frac{7 - 22}{2}}{12} \][/tex]
[tex]\[ m = \frac{\frac{-15}{2}}{12} \][/tex]
[tex]\[ m = \frac{-15}{2} \times \frac{1}{12} \][/tex]
[tex]\[ m = -\frac{15}{24} \][/tex]
[tex]\[ m = -\frac{5}{8} \][/tex]
So, the slope [tex]\(m\)[/tex] of the line is [tex]\(-\frac{5}{8}\)[/tex].
### Step 2: Calculate the Y-Intercept ([tex]\(b\)[/tex])
Using the slope-intercept form [tex]\(y = mx + b\)[/tex], we substitute one of the points and the slope to solve for [tex]\(b\)[/tex].
Using the point [tex]\((-8, 11)\)[/tex] and the slope [tex]\(m = -\frac{5}{8}\)[/tex]:
[tex]\[ y = mx + b \][/tex]
[tex]\[ 11 = -\frac{5}{8}(-8) + b \][/tex]
[tex]\[ 11 = \frac{40}{8} + b \][/tex]
[tex]\[ 11 = 5 + b \][/tex]
[tex]\[ b = 11 - 5 \][/tex]
[tex]\[ b = 6 \][/tex]
So, the y-intercept [tex]\(b\)[/tex] is 6.
### Step 3: Form the Equation of the Line
Now that we have the slope and the y-intercept, we can write the equation of the line:
[tex]\[ y = -\frac{5}{8}x + 6 \][/tex]
### Step 4: Match the Equation with the Given Choices
We need to check which of the provided equations match our derived equation:
1. [tex]\( y = -\frac{5}{8}x + 6 \)[/tex]
2. [tex]\( y = -\frac{5}{8}x + 16 \)[/tex]
3. [tex]\( y = -\frac{15}{2}x - 49 \)[/tex]
4. [tex]\( y = -\frac{15}{2}x + 71 \)[/tex]
From the calculations, it is clear that the equation:
[tex]\[ y = -\frac{5}{8}x + 6 \][/tex]
matches the equation we derived.
Thus, the correct equation representing the line passing through [tex]\((-8, 11)\)[/tex] and [tex]\(\left(4, \frac{7}{2}\right)\)[/tex] is:
[tex]\[ y = -\frac{5}{8}x + 6 \][/tex]
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