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Sagot :
To find the equation of the line that passes through the points (-2, -4) and (-5, -1), we will follow these steps:
1. Calculate the slope of the line.
2. Use the point-slope form to write the equation of the line.
3. Convert the point-slope form to the slope-intercept form.
4. Convert the slope-intercept form to the standard form.
### Step 1: Calculate the Slope
The formula for the slope (m) between 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]
Substituting the given points [tex]\((-2, -4)\)[/tex] and [tex]\((-5, -1)\)[/tex]:
[tex]\[ m = \frac{-1 - (-4)}{-5 - (-2)} \][/tex]
[tex]\[ m = \frac{-1 + 4}{-5 + 2} \][/tex]
[tex]\[ m = \frac{3}{-3} \][/tex]
[tex]\[ m = -1 \][/tex]
So the slope of the line is [tex]\(-1\)[/tex].
### Step 2: Point-Slope Form
The point-slope form of a line with slope [tex]\(m\)[/tex] going through the point [tex]\((x_1, y_1)\)[/tex] is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using the point [tex]\((-2, -4)\)[/tex] and the slope [tex]\(-1\)[/tex]:
[tex]\[ y - (-4) = -1(x - (-2)) \][/tex]
[tex]\[ y + 4 = -1(x + 2) \][/tex]
Thus, the point-slope form of the line is:
[tex]\[ y + 4 = -1(x + 2) \][/tex]
### Step 3: Slope-Intercept Form
The slope-intercept form of a line is:
[tex]\[ y = mx + b \][/tex]
We start from the point-slope form and solve for [tex]\(y\)[/tex]:
[tex]\[ y + 4 = -1(x + 2) \][/tex]
[tex]\[ y + 4 = -1x - 2 \][/tex]
[tex]\[ y = -1x - 2 - 4 \][/tex]
[tex]\[ y = -1x - 6 \][/tex]
Thus, the slope-intercept form of the line is:
[tex]\[ y = -1x - 6 \][/tex]
### Step 4: Standard Form
The standard form of a line is:
[tex]\[ Ax + By = C \][/tex]
Starting from the slope-intercept form [tex]\(y = -1x - 6\)[/tex], we need to rearrange it to match the standard form:
[tex]\[ y = -1x - 6 \][/tex]
First, we add [tex]\(1x\)[/tex] to both sides:
[tex]\[ 1x + y = -6 \][/tex]
To match the standard form [tex]\(Ax + By = C\)[/tex], we identify [tex]\(A = 1\)[/tex], [tex]\(B = 1\)[/tex], and [tex]\(C = -6\)[/tex].
Thus, the standard form of the line is:
[tex]\[ 1x + 1y = -6 \][/tex]
So, we have the following equations for the line that passes through [tex]\((-2, -4)\)[/tex] and [tex]\((-5, -1)\)[/tex]:
1. Point-slope form: [tex]\( y + 4 = -1(x + 2) \)[/tex]
2. Slope-intercept form: [tex]\( y = -1x - 6 \)[/tex]
3. Standard form: [tex]\( 1x + 1y = -6 \)[/tex]
1. Calculate the slope of the line.
2. Use the point-slope form to write the equation of the line.
3. Convert the point-slope form to the slope-intercept form.
4. Convert the slope-intercept form to the standard form.
### Step 1: Calculate the Slope
The formula for the slope (m) between 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]
Substituting the given points [tex]\((-2, -4)\)[/tex] and [tex]\((-5, -1)\)[/tex]:
[tex]\[ m = \frac{-1 - (-4)}{-5 - (-2)} \][/tex]
[tex]\[ m = \frac{-1 + 4}{-5 + 2} \][/tex]
[tex]\[ m = \frac{3}{-3} \][/tex]
[tex]\[ m = -1 \][/tex]
So the slope of the line is [tex]\(-1\)[/tex].
### Step 2: Point-Slope Form
The point-slope form of a line with slope [tex]\(m\)[/tex] going through the point [tex]\((x_1, y_1)\)[/tex] is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using the point [tex]\((-2, -4)\)[/tex] and the slope [tex]\(-1\)[/tex]:
[tex]\[ y - (-4) = -1(x - (-2)) \][/tex]
[tex]\[ y + 4 = -1(x + 2) \][/tex]
Thus, the point-slope form of the line is:
[tex]\[ y + 4 = -1(x + 2) \][/tex]
### Step 3: Slope-Intercept Form
The slope-intercept form of a line is:
[tex]\[ y = mx + b \][/tex]
We start from the point-slope form and solve for [tex]\(y\)[/tex]:
[tex]\[ y + 4 = -1(x + 2) \][/tex]
[tex]\[ y + 4 = -1x - 2 \][/tex]
[tex]\[ y = -1x - 2 - 4 \][/tex]
[tex]\[ y = -1x - 6 \][/tex]
Thus, the slope-intercept form of the line is:
[tex]\[ y = -1x - 6 \][/tex]
### Step 4: Standard Form
The standard form of a line is:
[tex]\[ Ax + By = C \][/tex]
Starting from the slope-intercept form [tex]\(y = -1x - 6\)[/tex], we need to rearrange it to match the standard form:
[tex]\[ y = -1x - 6 \][/tex]
First, we add [tex]\(1x\)[/tex] to both sides:
[tex]\[ 1x + y = -6 \][/tex]
To match the standard form [tex]\(Ax + By = C\)[/tex], we identify [tex]\(A = 1\)[/tex], [tex]\(B = 1\)[/tex], and [tex]\(C = -6\)[/tex].
Thus, the standard form of the line is:
[tex]\[ 1x + 1y = -6 \][/tex]
So, we have the following equations for the line that passes through [tex]\((-2, -4)\)[/tex] and [tex]\((-5, -1)\)[/tex]:
1. Point-slope form: [tex]\( y + 4 = -1(x + 2) \)[/tex]
2. Slope-intercept form: [tex]\( y = -1x - 6 \)[/tex]
3. Standard form: [tex]\( 1x + 1y = -6 \)[/tex]
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