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To determine which equation could make the system of equations have no solution when paired with [tex]\(y = 8x + 7\)[/tex], we need to analyze the properties of parallel lines. Two lines will have no points of intersection if they are parallel but have different y-intercepts.
First, let's outline the equation given:
- [tex]\( y = 8x + 7 \)[/tex]
This line has a slope [tex]\( m = 8 \)[/tex].
Next, let's consider each of the options given and determine their slopes and whether they are parallel to the given line.
1. Option 1: [tex]\( 2y = 16x + 14 \)[/tex]
- Simplify this equation by dividing everything by 2:
[tex]\[ y = 8x + 7 \][/tex]
- This equation has a slope of [tex]\( 8 \)[/tex] and a y-intercept of [tex]\( 7 \)[/tex], making it identical to the given equation. Since both lines are the same, they intersect at every point on the line, hence they are not parallel and do not contribute to a system with no solution.
2. Option 2: [tex]\( y = 8x - 7 \)[/tex]
- This equation is already in slope-intercept form [tex]\( y = mx + b \)[/tex].
- The slope is [tex]\( 8 \)[/tex], which is the same as the slope of the given line.
- The y-intercept here is [tex]\( -7 \)[/tex], which is different from [tex]\( 7 \)[/tex] in the given equation.
- Since they have the same slope but different y-intercepts, these lines are parallel and will never intersect. This pair of lines creates a system of equations that has no solution.
3. Option 3: [tex]\( y = -8x + 7 \)[/tex]
- This equation has a slope of [tex]\( -8 \)[/tex], which is different from the slope of the given line.
- Because the slopes are different, these lines are not parallel and will intersect at some point, thus they cannot be used to create a system with no solution.
4. Option 4: [tex]\( 2y = -16x - 14 \)[/tex]
- Simplify this by dividing everything by 2:
[tex]\[ y = -8x - 7 \][/tex]
- This equation has a slope of [tex]\( -8 \)[/tex], which is different from the slope of the given line.
- Due to the difference in slopes, these lines are not parallel and will intersect, thus they do not contribute to a system with no solution.
Given this analysis, the other equation in the system of equations that would result in no solution when paired with [tex]\( y = 8x + 7 \)[/tex] is:
Option 2: [tex]\( y = 8x - 7 \)[/tex].
First, let's outline the equation given:
- [tex]\( y = 8x + 7 \)[/tex]
This line has a slope [tex]\( m = 8 \)[/tex].
Next, let's consider each of the options given and determine their slopes and whether they are parallel to the given line.
1. Option 1: [tex]\( 2y = 16x + 14 \)[/tex]
- Simplify this equation by dividing everything by 2:
[tex]\[ y = 8x + 7 \][/tex]
- This equation has a slope of [tex]\( 8 \)[/tex] and a y-intercept of [tex]\( 7 \)[/tex], making it identical to the given equation. Since both lines are the same, they intersect at every point on the line, hence they are not parallel and do not contribute to a system with no solution.
2. Option 2: [tex]\( y = 8x - 7 \)[/tex]
- This equation is already in slope-intercept form [tex]\( y = mx + b \)[/tex].
- The slope is [tex]\( 8 \)[/tex], which is the same as the slope of the given line.
- The y-intercept here is [tex]\( -7 \)[/tex], which is different from [tex]\( 7 \)[/tex] in the given equation.
- Since they have the same slope but different y-intercepts, these lines are parallel and will never intersect. This pair of lines creates a system of equations that has no solution.
3. Option 3: [tex]\( y = -8x + 7 \)[/tex]
- This equation has a slope of [tex]\( -8 \)[/tex], which is different from the slope of the given line.
- Because the slopes are different, these lines are not parallel and will intersect at some point, thus they cannot be used to create a system with no solution.
4. Option 4: [tex]\( 2y = -16x - 14 \)[/tex]
- Simplify this by dividing everything by 2:
[tex]\[ y = -8x - 7 \][/tex]
- This equation has a slope of [tex]\( -8 \)[/tex], which is different from the slope of the given line.
- Due to the difference in slopes, these lines are not parallel and will intersect, thus they do not contribute to a system with no solution.
Given this analysis, the other equation in the system of equations that would result in no solution when paired with [tex]\( y = 8x + 7 \)[/tex] is:
Option 2: [tex]\( y = 8x - 7 \)[/tex].
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