Answer :
To determine which values of [tex]\(m\)[/tex] and [tex]\(b\)[/tex] will create a system of linear equations with no solution, let's analyze the given system of equations step-by-step.
The given system of equations is:
[tex]\[ y = -3x + 5 \][/tex]
[tex]\[ y = mx + b \][/tex]
### Step-by-Step Solution:
1. Identify the Slope and Intercept of the Given Equation:
- The equation [tex]\( y = -3x + 5 \)[/tex] is already in slope-intercept form [tex]\( y = mx + b \)[/tex].
- Here, the slope ([tex]\( m \)[/tex]) is [tex]\(-3\)[/tex] and the y-intercept ([tex]\( b \)[/tex]) is [tex]\(5\)[/tex].
2. Conditions for No Solution:
- For the system of equations to have no solution, the lines represented by the equations must be parallel but have different y-intercepts.
- This means that both lines must have the same slope but different y-intercepts.
3. Analyze Each Option:
- We need to check each provided option to see if it meets the conditions for no solution: [tex]\( m = -3 \)[/tex] and [tex]\( b \neq 5 \)[/tex].
- Option 1: [tex]\( m = -3 \)[/tex] and [tex]\( b = -3 \)[/tex]
- Slope: [tex]\(-3\)[/tex]
- Y-intercept: [tex]\(-3\)[/tex]
- Slope is the same as the given equation ([tex]\( m = -3 \)[/tex]) and the y-intercept is different ([tex]\( b \neq 5 \)[/tex]).
- This option will create a system of linear equations with no solution.
- Option 2: [tex]\( m = 5 \)[/tex] and [tex]\( b = -3 \)[/tex]
- Slope: [tex]\(5\)[/tex]
- Y-intercept: [tex]\(-3\)[/tex]
- Slope is different from the given equation ([tex]\( m \neq -3 \)[/tex]).
- This option will not create a system of linear equations with no solution.
- Option 3: [tex]\( m = 3 \)[/tex] and [tex]\( b = 5 \)[/tex]
- Slope: [tex]\(3\)[/tex]
- Y-intercept: [tex]\(5\)[/tex]
- Slope is different from the given equation ([tex]\( m \neq -3 \)[/tex]).
- This option will not create a system of linear equations with no solution.
- Option 4: [tex]\( m = -3 \)[/tex] and [tex]\( b = 5 \)[/tex]
- Slope: [tex]\(-3\)[/tex]
- Y-intercept: [tex]\(5\)[/tex]
- Slope is the same as the given equation ([tex]\( m = -3 \)[/tex]) but the y-intercept is the same ([tex]\( b = 5 \)[/tex]).
- This option will create a system of linear equations with infinitely many solutions (not no solution).
### Conclusion:
Based on the analysis, the values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex] that will create a system of linear equations with no solution are:
[tex]\[ m = -3 \][/tex] and [tex]\[ b = -3 \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{1} \][/tex]
The given system of equations is:
[tex]\[ y = -3x + 5 \][/tex]
[tex]\[ y = mx + b \][/tex]
### Step-by-Step Solution:
1. Identify the Slope and Intercept of the Given Equation:
- The equation [tex]\( y = -3x + 5 \)[/tex] is already in slope-intercept form [tex]\( y = mx + b \)[/tex].
- Here, the slope ([tex]\( m \)[/tex]) is [tex]\(-3\)[/tex] and the y-intercept ([tex]\( b \)[/tex]) is [tex]\(5\)[/tex].
2. Conditions for No Solution:
- For the system of equations to have no solution, the lines represented by the equations must be parallel but have different y-intercepts.
- This means that both lines must have the same slope but different y-intercepts.
3. Analyze Each Option:
- We need to check each provided option to see if it meets the conditions for no solution: [tex]\( m = -3 \)[/tex] and [tex]\( b \neq 5 \)[/tex].
- Option 1: [tex]\( m = -3 \)[/tex] and [tex]\( b = -3 \)[/tex]
- Slope: [tex]\(-3\)[/tex]
- Y-intercept: [tex]\(-3\)[/tex]
- Slope is the same as the given equation ([tex]\( m = -3 \)[/tex]) and the y-intercept is different ([tex]\( b \neq 5 \)[/tex]).
- This option will create a system of linear equations with no solution.
- Option 2: [tex]\( m = 5 \)[/tex] and [tex]\( b = -3 \)[/tex]
- Slope: [tex]\(5\)[/tex]
- Y-intercept: [tex]\(-3\)[/tex]
- Slope is different from the given equation ([tex]\( m \neq -3 \)[/tex]).
- This option will not create a system of linear equations with no solution.
- Option 3: [tex]\( m = 3 \)[/tex] and [tex]\( b = 5 \)[/tex]
- Slope: [tex]\(3\)[/tex]
- Y-intercept: [tex]\(5\)[/tex]
- Slope is different from the given equation ([tex]\( m \neq -3 \)[/tex]).
- This option will not create a system of linear equations with no solution.
- Option 4: [tex]\( m = -3 \)[/tex] and [tex]\( b = 5 \)[/tex]
- Slope: [tex]\(-3\)[/tex]
- Y-intercept: [tex]\(5\)[/tex]
- Slope is the same as the given equation ([tex]\( m = -3 \)[/tex]) but the y-intercept is the same ([tex]\( b = 5 \)[/tex]).
- This option will create a system of linear equations with infinitely many solutions (not no solution).
### Conclusion:
Based on the analysis, the values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex] that will create a system of linear equations with no solution are:
[tex]\[ m = -3 \][/tex] and [tex]\[ b = -3 \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{1} \][/tex]
