Answer :
Let's analyze each given situation step-by-step to verify the results for different changes and values in the line with a slope of [tex]\(2\)[/tex].
### 1. If [tex]\( x \)[/tex] decreases by 4:
The slope formula is given by:
[tex]\[ \text{slope} = \frac{\Delta y}{\Delta x} \][/tex]
Here, the slope is [tex]\(2\)[/tex]. Therefore, if [tex]\( x \)[/tex] decreases by [tex]\(4\)[/tex]:
[tex]\[ \Delta x = -4 \][/tex]
[tex]\[ \Delta y = \text{slope} \times \Delta x = 2 \times (-4) = -8 \][/tex]
So, if [tex]\( x \)[/tex] decreases by [tex]\(4\)[/tex], then [tex]\( y \)[/tex] decreases by [tex]\(8\)[/tex].
### 2. If [tex]\( x = -4 \)[/tex]:
Using the equation of the line, [tex]\( y = mx \)[/tex], where [tex]\( m \)[/tex] is the slope:
[tex]\[ y = 2 \times (-4) = -8 \][/tex]
So, if [tex]\( x = -4 \)[/tex], then [tex]\( y = -8 \)[/tex].
### 3. If [tex]\( y \)[/tex] decreases by 4:
Rewriting the slope formula:
[tex]\[ \text{slope} = \frac{\Delta y}{\Delta x} \][/tex]
Given the slope [tex]\(2\)[/tex] and [tex]\( \Delta y = -4 \)[/tex]:
[tex]\[ 2 = \frac{-4}{\Delta x} \][/tex]
Solving for [tex]\( \Delta x \)[/tex]:
[tex]\[ \Delta x = \frac{-4}{2} = -2 \][/tex]
So, if [tex]\( y \)[/tex] decreases by [tex]\(4\)[/tex], then [tex]\( x \)[/tex] decreases by [tex]\(2\)[/tex].
### 4. If [tex]\( x = -4 \)[/tex] and [tex]\( y = -8 \)[/tex]:
Substitute these values into the line equation [tex]\( y = mx \)[/tex]:
[tex]\[ y = 2x \][/tex]
[tex]\[ -8 = 2 \times (-4) = -8 \][/tex]
The equation holds true, confirming that the line’s equation is consistent with these values. Therefore, the point [tex]\((-4, -8)\)[/tex] lies on the line.
### 5. If [tex]\( x \)[/tex] increases by 4:
Using the slope:
[tex]\[ \Delta x = 4 \][/tex]
Thus:
[tex]\[ \Delta y = \text{slope} \times \Delta x = 2 \times 4 = 8 \][/tex]
So, if [tex]\( x \)[/tex] increases by [tex]\(4\)[/tex], [tex]\( y \)[/tex] increases by [tex]\(8\)[/tex].
### Summary of results:
- If [tex]\( x \)[/tex] decreases by [tex]\(4\)[/tex], [tex]\( y \)[/tex] decreases by [tex]\(8\)[/tex].
- If [tex]\( x = -4\)[/tex], [tex]\( y = -8\)[/tex].
- If [tex]\( y \)[/tex] decreases by [tex]\(4\)[/tex], [tex]\( x \)[/tex] decreases by [tex]\(2\)[/tex].
- For [tex]\( x = -4 \)[/tex] and [tex]\( y = -8 \)[/tex], the equation of the line holds true.
- If [tex]\( x \)[/tex] increases by [tex]\(4\)[/tex], [tex]\( y \)[/tex] increases by [tex]\(8\)[/tex].
The numerical results from our step-by-step analysis match perfectly with the given values:
- [tex]\(x\)[/tex] decreases by [tex]\(4 \rightarrow y\)[/tex] decreases by [tex]\(8\)[/tex].
- [tex]\(x = -4 \rightarrow y = -8\)[/tex].
- [tex]\(y\)[/tex] decreases by [tex]\(4 \rightarrow x\)[/tex] decreases by [tex]\(2\)[/tex].
- The equation of the line [tex]\(y = 2x\)[/tex] holds true for point [tex]\((-4, -8)\)[/tex].
- [tex]\(x\)[/tex] increases by [tex]\(4 \rightarrow y\)[/tex] increases by [tex]\(8\)[/tex].
Therefore, all statements provided are correct.
### 1. If [tex]\( x \)[/tex] decreases by 4:
The slope formula is given by:
[tex]\[ \text{slope} = \frac{\Delta y}{\Delta x} \][/tex]
Here, the slope is [tex]\(2\)[/tex]. Therefore, if [tex]\( x \)[/tex] decreases by [tex]\(4\)[/tex]:
[tex]\[ \Delta x = -4 \][/tex]
[tex]\[ \Delta y = \text{slope} \times \Delta x = 2 \times (-4) = -8 \][/tex]
So, if [tex]\( x \)[/tex] decreases by [tex]\(4\)[/tex], then [tex]\( y \)[/tex] decreases by [tex]\(8\)[/tex].
### 2. If [tex]\( x = -4 \)[/tex]:
Using the equation of the line, [tex]\( y = mx \)[/tex], where [tex]\( m \)[/tex] is the slope:
[tex]\[ y = 2 \times (-4) = -8 \][/tex]
So, if [tex]\( x = -4 \)[/tex], then [tex]\( y = -8 \)[/tex].
### 3. If [tex]\( y \)[/tex] decreases by 4:
Rewriting the slope formula:
[tex]\[ \text{slope} = \frac{\Delta y}{\Delta x} \][/tex]
Given the slope [tex]\(2\)[/tex] and [tex]\( \Delta y = -4 \)[/tex]:
[tex]\[ 2 = \frac{-4}{\Delta x} \][/tex]
Solving for [tex]\( \Delta x \)[/tex]:
[tex]\[ \Delta x = \frac{-4}{2} = -2 \][/tex]
So, if [tex]\( y \)[/tex] decreases by [tex]\(4\)[/tex], then [tex]\( x \)[/tex] decreases by [tex]\(2\)[/tex].
### 4. If [tex]\( x = -4 \)[/tex] and [tex]\( y = -8 \)[/tex]:
Substitute these values into the line equation [tex]\( y = mx \)[/tex]:
[tex]\[ y = 2x \][/tex]
[tex]\[ -8 = 2 \times (-4) = -8 \][/tex]
The equation holds true, confirming that the line’s equation is consistent with these values. Therefore, the point [tex]\((-4, -8)\)[/tex] lies on the line.
### 5. If [tex]\( x \)[/tex] increases by 4:
Using the slope:
[tex]\[ \Delta x = 4 \][/tex]
Thus:
[tex]\[ \Delta y = \text{slope} \times \Delta x = 2 \times 4 = 8 \][/tex]
So, if [tex]\( x \)[/tex] increases by [tex]\(4\)[/tex], [tex]\( y \)[/tex] increases by [tex]\(8\)[/tex].
### Summary of results:
- If [tex]\( x \)[/tex] decreases by [tex]\(4\)[/tex], [tex]\( y \)[/tex] decreases by [tex]\(8\)[/tex].
- If [tex]\( x = -4\)[/tex], [tex]\( y = -8\)[/tex].
- If [tex]\( y \)[/tex] decreases by [tex]\(4\)[/tex], [tex]\( x \)[/tex] decreases by [tex]\(2\)[/tex].
- For [tex]\( x = -4 \)[/tex] and [tex]\( y = -8 \)[/tex], the equation of the line holds true.
- If [tex]\( x \)[/tex] increases by [tex]\(4\)[/tex], [tex]\( y \)[/tex] increases by [tex]\(8\)[/tex].
The numerical results from our step-by-step analysis match perfectly with the given values:
- [tex]\(x\)[/tex] decreases by [tex]\(4 \rightarrow y\)[/tex] decreases by [tex]\(8\)[/tex].
- [tex]\(x = -4 \rightarrow y = -8\)[/tex].
- [tex]\(y\)[/tex] decreases by [tex]\(4 \rightarrow x\)[/tex] decreases by [tex]\(2\)[/tex].
- The equation of the line [tex]\(y = 2x\)[/tex] holds true for point [tex]\((-4, -8)\)[/tex].
- [tex]\(x\)[/tex] increases by [tex]\(4 \rightarrow y\)[/tex] increases by [tex]\(8\)[/tex].
Therefore, all statements provided are correct.
