Answer :
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 the conditions under which a system of linear equations has no solution.
A system of linear equations has no solution when the lines represented by the equations are parallel but not identical. For the lines to be parallel, their slopes must be equal. For them not to be identical, their intercepts must be different.
Consider the given equations:
1. [tex]\( y = -3x + 5 \)[/tex]
2. [tex]\( y = mx + b \)[/tex]
For the two lines to be parallel:
- The slopes ([tex]\( m \)[/tex]) must be equal. So, [tex]\( m \)[/tex] must be [tex]\(-3\)[/tex].
For the two lines not to be identical:
- The y-intercepts ([tex]\( b \)[/tex]) must be different. So, [tex]\( b \)[/tex] must not be [tex]\( 5 \)[/tex].
We are given four options:
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]
Let's analyze each option:
1. [tex]\( m = -3 \)[/tex] and [tex]\( b = -3 \)[/tex]:
- The slopes are the same, [tex]\( m = -3 \)[/tex].
- The intercepts are different, [tex]\( b \neq 5 \)[/tex].
- Thus, the lines are parallel and not identical. This condition creates a system of linear equations with no solution.
2. [tex]\( m = 5 \)[/tex] and [tex]\( b = -3 \)[/tex]:
- The slopes are different, [tex]\( m \neq -3 \)[/tex].
- This means the lines are not parallel and therefore could intersect. This condition does not create a system of linear equations with no solution.
3. [tex]\( m = 3 \)[/tex] and [tex]\( b = 5 \)[/tex]:
- The slopes are different, [tex]\( m \neq -3 \)[/tex].
- This means the lines are not parallel and therefore could intersect. This condition does not create a system of linear equations with no solution.
4. [tex]\( m = -3 \)[/tex] and [tex]\( b = 5 \)[/tex]:
- The slopes are the same, [tex]\( m = -3 \)[/tex].
- The intercepts are the same, [tex]\( b = 5 \)[/tex].
- Thus, the lines are identical and have infinitely many solutions. This condition does not create a system of linear equations with no solution.
Therefore, the values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex] that create a system of linear equations with no solution are:
[tex]\[ m = -3 \text{ and } b = -3 \][/tex]
So, the correct option is:
[tex]\[ m = -3 \text{ and } b = -3 \][/tex]
Thus, the system of linear equations with no solution is given when:
[tex]\[ \boxed{1} \][/tex]
A system of linear equations has no solution when the lines represented by the equations are parallel but not identical. For the lines to be parallel, their slopes must be equal. For them not to be identical, their intercepts must be different.
Consider the given equations:
1. [tex]\( y = -3x + 5 \)[/tex]
2. [tex]\( y = mx + b \)[/tex]
For the two lines to be parallel:
- The slopes ([tex]\( m \)[/tex]) must be equal. So, [tex]\( m \)[/tex] must be [tex]\(-3\)[/tex].
For the two lines not to be identical:
- The y-intercepts ([tex]\( b \)[/tex]) must be different. So, [tex]\( b \)[/tex] must not be [tex]\( 5 \)[/tex].
We are given four options:
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]
Let's analyze each option:
1. [tex]\( m = -3 \)[/tex] and [tex]\( b = -3 \)[/tex]:
- The slopes are the same, [tex]\( m = -3 \)[/tex].
- The intercepts are different, [tex]\( b \neq 5 \)[/tex].
- Thus, the lines are parallel and not identical. This condition creates a system of linear equations with no solution.
2. [tex]\( m = 5 \)[/tex] and [tex]\( b = -3 \)[/tex]:
- The slopes are different, [tex]\( m \neq -3 \)[/tex].
- This means the lines are not parallel and therefore could intersect. This condition does not create a system of linear equations with no solution.
3. [tex]\( m = 3 \)[/tex] and [tex]\( b = 5 \)[/tex]:
- The slopes are different, [tex]\( m \neq -3 \)[/tex].
- This means the lines are not parallel and therefore could intersect. This condition does not create a system of linear equations with no solution.
4. [tex]\( m = -3 \)[/tex] and [tex]\( b = 5 \)[/tex]:
- The slopes are the same, [tex]\( m = -3 \)[/tex].
- The intercepts are the same, [tex]\( b = 5 \)[/tex].
- Thus, the lines are identical and have infinitely many solutions. This condition does not create a system of linear equations with no solution.
Therefore, the values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex] that create a system of linear equations with no solution are:
[tex]\[ m = -3 \text{ and } b = -3 \][/tex]
So, the correct option is:
[tex]\[ m = -3 \text{ and } b = -3 \][/tex]
Thus, the system of linear equations with no solution is given when:
[tex]\[ \boxed{1} \][/tex]
