Let's analyze the given system of linear equations:
1. [tex]\( y = 3x + 5 \)[/tex]
2. [tex]\( y = ax + b \)[/tex]
To determine values for [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that result in either an inconsistent system or a consistent and dependent system, we should consider their geometric interpretations.
### Inconsistent System
A system of equations is inconsistent if there are no solutions. For linear equations, this occurs when the lines represented by the equations are parallel but not coincident (they do not overlap).
Two lines are parallel if their slopes are the same. The slope of the first equation [tex]\( y = 3x + 5 \)[/tex] is 3. Therefore, to be inconsistent, the second equation must have the same slope. This means [tex]\(a\)[/tex] must be equal to 3. However, for the lines to be parallel and not coincident (i.e., to have different [tex]\( y \)[/tex]-intercepts), the [tex]\( y \)[/tex]-intercepts must differ. The [tex]\( y \)[/tex]-intercept of the second line is [tex]\(b\)[/tex].
Therefore, for the system to be inconsistent:
- [tex]\(a = 3\)[/tex]
- [tex]\(b\)[/tex] must be any value except 5
### Consistent and Dependent System
A system of equations is consistent and dependent if there are infinitely many solutions. In other words, the two equations represent the same line, meaning they overlap completely.
For the two equations to be identical, their slopes and [tex]\( y \)[/tex]-intercepts must be the same. The slope of the first equation [tex]\( y = 3x + 5 \)[/tex] is 3 and the [tex]\( y \)[/tex]-intercept is 5. So the second equation must also have a slope of 3 and a [tex]\( y \)[/tex]-intercept of 5.
Therefore, for the system to be consistent and dependent:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = 5\)[/tex]
In summary:
- For the system to be inconsistent: [tex]\( a = 3 \)[/tex] and [tex]\( b \neq 5 \)[/tex]
- For the system to be consistent and dependent: [tex]\( a = 3 \)[/tex] and [tex]\( b = 5 \)[/tex]
