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Sagot :
To determine which values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex] will create a system of linear equations with no solution, we need to understand what it means for two linear equations to have no solution.
For the given system of linear equations:
[tex]\[ \begin{array}{l} y = -3x + 5 \\ y = mx + b \end{array} \][/tex]
The lines will have no solution if they are parallel but have different y-intercepts. Two lines are parallel if they have the same slope. Therefore, for the system to have no solution:
1. The slopes of both lines [tex]\(m\)[/tex] must be the same.
2. The y-intercepts [tex]\(b\)[/tex] must be different.
The slope of the first line, [tex]\(y = -3x + 5\)[/tex], is [tex]\(-3\)[/tex]. Thus, for the second line [tex]\( y = mx + b \)[/tex] to be parallel to the first line, [tex]\( m \)[/tex] must also be [tex]\(-3\)[/tex].
However, for the system to have no solution, the y-intercept of the second line, [tex]\( b \)[/tex], must be different from the y-intercept of the first line which is [tex]\(5\)[/tex].
Let's analyze the given choices:
1. [tex]\( m = -3 \)[/tex] and [tex]\( b = -3 \)[/tex]
2. [tex]\( m = 5 \)[/tex] and [tex]\( b = -3 \)[/tex]
3. [tex]\( m = 3 \)[/tex] and [tex]\( b = 5 \)[/tex]
4. [tex]\( m = -3 \)[/tex] and [tex]\( b = 5 \)[/tex]
- For the first choice, [tex]\( m = -3 \)[/tex] and [tex]\( b = -3 \)[/tex]:
The slope is [tex]\(-3\)[/tex] which is the same as the first line.
The y-intercept is [tex]\(-3\)[/tex] which is different from the first line's y-intercept of [tex]\(5\)[/tex].
So, these lines are parallel and have different y-intercepts; hence, there is no solution.
- For the second choice, [tex]\( m = 5 \)[/tex] and [tex]\( b = -3 \)[/tex]:
The slope is [tex]\(5\)[/tex] which is different from [tex]\(-3\)[/tex].
Since the slopes are different, these lines will intersect at some point, meaning the system will have a solution.
- For the third choice, [tex]\( m = 3 \)[/tex] and [tex]\( b = 5 \)[/tex]:
The slope is [tex]\(3\)[/tex] which is different from [tex]\(-3\)[/tex].
Since the slopes are different, these lines will intersect at some point, meaning the system will have a solution.
- For the fourth choice, [tex]\( m = -3 \)[/tex] and [tex]\( b = 5 \)[/tex]:
The slope is [tex]\(-3\)[/tex] which is the same as the first line.
The y-intercept is [tex]\(5\)[/tex] which is the same as the first line's y-intercept.
So, these lines are not only parallel but also coincident, meaning they have infinitely many solutions (every point on one line is also on the other line).
Among the choices given, only the first choice, [tex]\( m = -3 \)[/tex] and [tex]\( b = -3 \)[/tex], satisfies the condition of having the same slope but a different y-intercept. Therefore, it results in a system of equations with no solution.
The correct values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex] that satisfy the condition of no solution are:
[tex]\[ \boxed{m = -3 \text{ and } b = -3} \][/tex]
So the selection for the given question is the first choice.
For the given system of linear equations:
[tex]\[ \begin{array}{l} y = -3x + 5 \\ y = mx + b \end{array} \][/tex]
The lines will have no solution if they are parallel but have different y-intercepts. Two lines are parallel if they have the same slope. Therefore, for the system to have no solution:
1. The slopes of both lines [tex]\(m\)[/tex] must be the same.
2. The y-intercepts [tex]\(b\)[/tex] must be different.
The slope of the first line, [tex]\(y = -3x + 5\)[/tex], is [tex]\(-3\)[/tex]. Thus, for the second line [tex]\( y = mx + b \)[/tex] to be parallel to the first line, [tex]\( m \)[/tex] must also be [tex]\(-3\)[/tex].
However, for the system to have no solution, the y-intercept of the second line, [tex]\( b \)[/tex], must be different from the y-intercept of the first line which is [tex]\(5\)[/tex].
Let's analyze the given choices:
1. [tex]\( m = -3 \)[/tex] and [tex]\( b = -3 \)[/tex]
2. [tex]\( m = 5 \)[/tex] and [tex]\( b = -3 \)[/tex]
3. [tex]\( m = 3 \)[/tex] and [tex]\( b = 5 \)[/tex]
4. [tex]\( m = -3 \)[/tex] and [tex]\( b = 5 \)[/tex]
- For the first choice, [tex]\( m = -3 \)[/tex] and [tex]\( b = -3 \)[/tex]:
The slope is [tex]\(-3\)[/tex] which is the same as the first line.
The y-intercept is [tex]\(-3\)[/tex] which is different from the first line's y-intercept of [tex]\(5\)[/tex].
So, these lines are parallel and have different y-intercepts; hence, there is no solution.
- For the second choice, [tex]\( m = 5 \)[/tex] and [tex]\( b = -3 \)[/tex]:
The slope is [tex]\(5\)[/tex] which is different from [tex]\(-3\)[/tex].
Since the slopes are different, these lines will intersect at some point, meaning the system will have a solution.
- For the third choice, [tex]\( m = 3 \)[/tex] and [tex]\( b = 5 \)[/tex]:
The slope is [tex]\(3\)[/tex] which is different from [tex]\(-3\)[/tex].
Since the slopes are different, these lines will intersect at some point, meaning the system will have a solution.
- For the fourth choice, [tex]\( m = -3 \)[/tex] and [tex]\( b = 5 \)[/tex]:
The slope is [tex]\(-3\)[/tex] which is the same as the first line.
The y-intercept is [tex]\(5\)[/tex] which is the same as the first line's y-intercept.
So, these lines are not only parallel but also coincident, meaning they have infinitely many solutions (every point on one line is also on the other line).
Among the choices given, only the first choice, [tex]\( m = -3 \)[/tex] and [tex]\( b = -3 \)[/tex], satisfies the condition of having the same slope but a different y-intercept. Therefore, it results in a system of equations with no solution.
The correct values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex] that satisfy the condition of no solution are:
[tex]\[ \boxed{m = -3 \text{ and } b = -3} \][/tex]
So the selection for the given question is the first choice.
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