Answer :
To determine which of the given equations would result in a system of equations that has no solution with the equation [tex]\( y = 8x + 7 \)[/tex], we need to identify an equation that is parallel to [tex]\( y = 8x + 7 \)[/tex]. Parallel lines have the same slope but different y-intercepts.
Let’s analyze each of the given equations to find their slopes and y-intercepts:
1. [tex]\( 2y = 16x + 14 \)[/tex]
- First, we rearrange it to the slope-intercept form [tex]\( y = mx + b \)[/tex].
- Divide both sides by 2:
[tex]\[ y = 8x + 7 \][/tex]
- This equation has a slope of 8 and a y-intercept of 7, which is exactly the same as [tex]\( y = 8x + 7 \)[/tex]. Since they are the same line, they do not count as parallel lines.
2. [tex]\( y = 8x - 7 \)[/tex]
- This equation is already in slope-intercept form: [tex]\( y = 8x - 7 \)[/tex].
- It has a slope of 8 and a y-intercept of -7.
- The slope is the same as [tex]\( y = 8x + 7 \)[/tex], but the y-intercepts are different.
- Therefore, this line is parallel to [tex]\( y = 8x + 7 \)[/tex], indicating that the system has no solution when paired with [tex]\( y = 8x + 7 \)[/tex].
3. [tex]\( y = -8x + 7 \)[/tex]
- This equation is already in slope-intercept form: [tex]\( y = -8x + 7 \)[/tex].
- It has a slope of -8 and a y-intercept of 7.
- The slope is different from that of [tex]\( y = 8x + 7 \)[/tex], so this line is not parallel to it.
4. [tex]\( 2y = -16x - 14 \)[/tex]
- We need to rearrange it to the slope-intercept form.
- Divide both sides by 2:
[tex]\[ y = -8x - 7 \][/tex]
- This equation has a slope of -8 and a y-intercept of -7.
- The slope is different from that of [tex]\( y = 8x + 7 \)[/tex], so this line is not parallel to it.
From the analysis above, the equation that makes the system have no solution with [tex]\( y = 8x + 7 \)[/tex] is [tex]\( y = 8x - 7 \)[/tex], as it is parallel to [tex]\( y = 8x + 7 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{y = 8x - 7} \][/tex]
Let’s analyze each of the given equations to find their slopes and y-intercepts:
1. [tex]\( 2y = 16x + 14 \)[/tex]
- First, we rearrange it to the slope-intercept form [tex]\( y = mx + b \)[/tex].
- Divide both sides by 2:
[tex]\[ y = 8x + 7 \][/tex]
- This equation has a slope of 8 and a y-intercept of 7, which is exactly the same as [tex]\( y = 8x + 7 \)[/tex]. Since they are the same line, they do not count as parallel lines.
2. [tex]\( y = 8x - 7 \)[/tex]
- This equation is already in slope-intercept form: [tex]\( y = 8x - 7 \)[/tex].
- It has a slope of 8 and a y-intercept of -7.
- The slope is the same as [tex]\( y = 8x + 7 \)[/tex], but the y-intercepts are different.
- Therefore, this line is parallel to [tex]\( y = 8x + 7 \)[/tex], indicating that the system has no solution when paired with [tex]\( y = 8x + 7 \)[/tex].
3. [tex]\( y = -8x + 7 \)[/tex]
- This equation is already in slope-intercept form: [tex]\( y = -8x + 7 \)[/tex].
- It has a slope of -8 and a y-intercept of 7.
- The slope is different from that of [tex]\( y = 8x + 7 \)[/tex], so this line is not parallel to it.
4. [tex]\( 2y = -16x - 14 \)[/tex]
- We need to rearrange it to the slope-intercept form.
- Divide both sides by 2:
[tex]\[ y = -8x - 7 \][/tex]
- This equation has a slope of -8 and a y-intercept of -7.
- The slope is different from that of [tex]\( y = 8x + 7 \)[/tex], so this line is not parallel to it.
From the analysis above, the equation that makes the system have no solution with [tex]\( y = 8x + 7 \)[/tex] is [tex]\( y = 8x - 7 \)[/tex], as it is parallel to [tex]\( y = 8x + 7 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{y = 8x - 7} \][/tex]
