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Sagot :
Let's analyze the line passing through the points [tex]\((-2, 0)\)[/tex] and [tex]\((2, -4)\)[/tex] step by step and evaluate each of the given statements.
1. Calculate the slope of the line:
The formula for 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:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the given points [tex]\((-2, 0)\)[/tex] and [tex]\((2, -4)\)[/tex]:
[tex]\[ m = \frac{-4 - 0}{2 - (-2)} = \frac{-4}{4} = -1 \][/tex]
So, the slope of the line is [tex]\(-1\)[/tex].
2. Calculate the y-intercept of the line:
The y-intercept [tex]\( b \)[/tex] can be found using the equation of the line in slope-intercept form [tex]\( y = mx + b \)[/tex]. Substituting one of the points [tex]\((x_1, y_1)\)[/tex] and the slope [tex]\( m \)[/tex]:
[tex]\[ y_1 = m x_1 + b \][/tex]
Using [tex]\((x_1, y_1) = (-2, 0)\)[/tex] and [tex]\( m = -1\)[/tex]:
[tex]\[ 0 = -1 \cdot (-2) + b \implies 0 = 2 + b \implies b = -2 \][/tex]
So, the y-intercept of the line is [tex]\(-2\)[/tex].
3. Form the equation of the line:
Now that we have the slope [tex]\( m = -1 \)[/tex] and the y-intercept [tex]\( b = -2 \)[/tex], we can write the equation of the line:
[tex]\[ y = -1x - 2 \][/tex]
Or simply:
[tex]\[ y = -x - 2 \][/tex]
4. Determine the intersection with the x-axis:
To find the x-intercept, we set [tex]\( y = 0 \)[/tex] in the equation of the line and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = -x - 2 \implies x = -2 \][/tex]
So, the line intersects the x-axis at [tex]\((-2, 0)\)[/tex].
Now, let’s evaluate each of the provided statements based on the results:
A. The slope of the line is 1.
- This is false. We calculated the slope to be [tex]\(-1\)[/tex].
B. The line intersects the y-axis at [tex]\((0, -2)\)[/tex].
- This is true. We determined the y-intercept to be [tex]\(-2\)[/tex], so the line intersects the y-axis at [tex]\((0, -2)\)[/tex].
C. The equation of the line is [tex]\( y = -x - 2 \)[/tex].
- This is true. We formed the equation of the line to be [tex]\( y = -x - 2 \)[/tex].
D. The line intersects the x-axis at [tex]\((-2, 0)\)[/tex].
- This is true. We determined the line intersects the x-axis at [tex]\((-2, 0)\)[/tex].
Therefore, the true statements are:
- B. The line intersects the y-axis at [tex]$(0, -2)$[/tex].
- C. The equation of the line is [tex]$y = -x - 2$[/tex].
- D. The line intersects the x-axis at [tex]$(-2, 0)$[/tex].
The correct selection is:
B, C, and D.
1. Calculate the slope of the line:
The formula for 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:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the given points [tex]\((-2, 0)\)[/tex] and [tex]\((2, -4)\)[/tex]:
[tex]\[ m = \frac{-4 - 0}{2 - (-2)} = \frac{-4}{4} = -1 \][/tex]
So, the slope of the line is [tex]\(-1\)[/tex].
2. Calculate the y-intercept of the line:
The y-intercept [tex]\( b \)[/tex] can be found using the equation of the line in slope-intercept form [tex]\( y = mx + b \)[/tex]. Substituting one of the points [tex]\((x_1, y_1)\)[/tex] and the slope [tex]\( m \)[/tex]:
[tex]\[ y_1 = m x_1 + b \][/tex]
Using [tex]\((x_1, y_1) = (-2, 0)\)[/tex] and [tex]\( m = -1\)[/tex]:
[tex]\[ 0 = -1 \cdot (-2) + b \implies 0 = 2 + b \implies b = -2 \][/tex]
So, the y-intercept of the line is [tex]\(-2\)[/tex].
3. Form the equation of the line:
Now that we have the slope [tex]\( m = -1 \)[/tex] and the y-intercept [tex]\( b = -2 \)[/tex], we can write the equation of the line:
[tex]\[ y = -1x - 2 \][/tex]
Or simply:
[tex]\[ y = -x - 2 \][/tex]
4. Determine the intersection with the x-axis:
To find the x-intercept, we set [tex]\( y = 0 \)[/tex] in the equation of the line and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = -x - 2 \implies x = -2 \][/tex]
So, the line intersects the x-axis at [tex]\((-2, 0)\)[/tex].
Now, let’s evaluate each of the provided statements based on the results:
A. The slope of the line is 1.
- This is false. We calculated the slope to be [tex]\(-1\)[/tex].
B. The line intersects the y-axis at [tex]\((0, -2)\)[/tex].
- This is true. We determined the y-intercept to be [tex]\(-2\)[/tex], so the line intersects the y-axis at [tex]\((0, -2)\)[/tex].
C. The equation of the line is [tex]\( y = -x - 2 \)[/tex].
- This is true. We formed the equation of the line to be [tex]\( y = -x - 2 \)[/tex].
D. The line intersects the x-axis at [tex]\((-2, 0)\)[/tex].
- This is true. We determined the line intersects the x-axis at [tex]\((-2, 0)\)[/tex].
Therefore, the true statements are:
- B. The line intersects the y-axis at [tex]$(0, -2)$[/tex].
- C. The equation of the line is [tex]$y = -x - 2$[/tex].
- D. The line intersects the x-axis at [tex]$(-2, 0)$[/tex].
The correct selection is:
B, C, and D.
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