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To determine which equation, when paired with [tex]\( y = 3x - 1 \)[/tex], creates a system with an infinite number of solutions, we need to understand a key concept: Systems with an infinite number of solutions actually represent the same line. This implies that the second equation must essentially be the same line as [tex]\( y = 3x - 1 \)[/tex], possibly written in a different but equivalent form.
We will now analyze each option to see if it can be rewritten as [tex]\( y = 3x - 1 \)[/tex]:
1. Option: [tex]\( y = 3x + 2 \)[/tex]
- This equation is already in the slope-intercept form [tex]\( y = mx + c \)[/tex], where the slope [tex]\( m \)[/tex] is 3, but the intercept is 2.
- Clearly, [tex]\( y = 3x + 2 \)[/tex] has a different intercept than [tex]\( y = 3x - 1 \)[/tex], so it represents a different line.
- This option cannot be correct because it does not match [tex]\( y = 3x - 1 \)[/tex].
2. Option: [tex]\( 3x - y = 2 \)[/tex]
- To check if this is equivalent to [tex]\( y = 3x - 1 \)[/tex], we need to solve for [tex]\( y \)[/tex]:
[tex]\[ 3x - y = 2 \implies -y = 2 - 3x \implies y = 3x - 2 \][/tex]
- Here, [tex]\( y = 3x - 2 \)[/tex], which also has the same slope as [tex]\( y = 3x - 1 \)[/tex], but the intercept is different.
- This equation is different from [tex]\( y = 3x - 1 \)[/tex], so it cannot be correct.
3. Option: [tex]\( 3x - y = 1 \)[/tex]
- Again, we solve for [tex]\( y \)[/tex]:
[tex]\[ 3x - y = 1 \implies -y = 1 - 3x \implies y = 3x - 1 \][/tex]
- Here, we find that [tex]\( y = 3x - 1 \)[/tex], which is exactly the same as the given equation.
- This option, therefore, represents the same line, meaning the system has an infinite number of solutions.
4. Option: [tex]\( 3x + y = 1 \)[/tex]
- Solving for [tex]\( y \)[/tex]:
[tex]\[ 3x + y = 1 \implies y = 1 - 3x \][/tex]
- This gives us [tex]\( y = -3x + 1 \)[/tex], which has a different slope from [tex]\( y = 3x - 1 \)[/tex].
- It is a completely different line, so this option cannot be correct.
Given all the analysis, the correct pairing that results in an infinite number of solutions is:
[tex]\[ 3x - y = 1 \][/tex]
Thus, the correct option is:
[tex]\[ 3x - y = 1 \][/tex]
We will now analyze each option to see if it can be rewritten as [tex]\( y = 3x - 1 \)[/tex]:
1. Option: [tex]\( y = 3x + 2 \)[/tex]
- This equation is already in the slope-intercept form [tex]\( y = mx + c \)[/tex], where the slope [tex]\( m \)[/tex] is 3, but the intercept is 2.
- Clearly, [tex]\( y = 3x + 2 \)[/tex] has a different intercept than [tex]\( y = 3x - 1 \)[/tex], so it represents a different line.
- This option cannot be correct because it does not match [tex]\( y = 3x - 1 \)[/tex].
2. Option: [tex]\( 3x - y = 2 \)[/tex]
- To check if this is equivalent to [tex]\( y = 3x - 1 \)[/tex], we need to solve for [tex]\( y \)[/tex]:
[tex]\[ 3x - y = 2 \implies -y = 2 - 3x \implies y = 3x - 2 \][/tex]
- Here, [tex]\( y = 3x - 2 \)[/tex], which also has the same slope as [tex]\( y = 3x - 1 \)[/tex], but the intercept is different.
- This equation is different from [tex]\( y = 3x - 1 \)[/tex], so it cannot be correct.
3. Option: [tex]\( 3x - y = 1 \)[/tex]
- Again, we solve for [tex]\( y \)[/tex]:
[tex]\[ 3x - y = 1 \implies -y = 1 - 3x \implies y = 3x - 1 \][/tex]
- Here, we find that [tex]\( y = 3x - 1 \)[/tex], which is exactly the same as the given equation.
- This option, therefore, represents the same line, meaning the system has an infinite number of solutions.
4. Option: [tex]\( 3x + y = 1 \)[/tex]
- Solving for [tex]\( y \)[/tex]:
[tex]\[ 3x + y = 1 \implies y = 1 - 3x \][/tex]
- This gives us [tex]\( y = -3x + 1 \)[/tex], which has a different slope from [tex]\( y = 3x - 1 \)[/tex].
- It is a completely different line, so this option cannot be correct.
Given all the analysis, the correct pairing that results in an infinite number of solutions is:
[tex]\[ 3x - y = 1 \][/tex]
Thus, the correct option is:
[tex]\[ 3x - y = 1 \][/tex]
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