To determine the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that make the system of equations either inconsistent or consistent and dependent, let's analyze the system of linear equations step-by-step:
Given:
1. [tex]\( y = 3x + 5 \)[/tex]
2. [tex]\( y = ax + b \)[/tex]
### Inconsistent System
For a system of linear equations to be inconsistent, the lines represented by the equations must be parallel but not overlapping. This happens if the slopes (the coefficients of [tex]\(x\)[/tex]) of the two lines are equal but their y-intercepts are different.
- Slope of the first line: In the equation [tex]\( y = 3x + 5 \)[/tex], the slope is [tex]\( 3 \)[/tex].
- Slope of the second line: In the equation [tex]\( y = ax + b \)[/tex], the slope is [tex]\( a \)[/tex].
For the lines to be parallel, the slopes must be equal:
[tex]\[ a = 3 \][/tex]
Next, for the lines to not overlap (which makes the system inconsistent), the y-intercepts must be different:
- The y-intercept of the first line is [tex]\( 5 \)[/tex].
- The y-intercept of the second line is [tex]\( b \)[/tex].
So, for inconsistency, [tex]\( b \)[/tex] must not be equal to [tex]\( 5 \)[/tex]:
[tex]\[ b \neq 5 \][/tex]
Thus, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that make the system inconsistent are:
[tex]\[ a = 3 \][/tex]
[tex]\[ b \text{ can be any value other than } 5 \][/tex]
### Consistent and Dependent System
For a system of linear equations to be consistent and dependent, the lines must be exactly the same. This means both their slopes and y-intercepts must be identical.
From the first equation ([tex]\( y = 3x + 5 \)[/tex]):
- The slope is [tex]\( 3 \)[/tex].
- The y-intercept is [tex]\( 5 \)[/tex].
For the second equation ([tex]\( y = ax + b \)[/tex]) to represent the same line:
- The slope [tex]\( a \)[/tex] must be [tex]\( 3 \)[/tex], and
- The y-intercept [tex]\( b \)[/tex] must be [tex]\( 5 \)[/tex].
So, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that make the system consistent and dependent are:
[tex]\[ a = 3 \][/tex]
[tex]\[ b = 5 \][/tex]
### Summary
- The system is inconsistent when [tex]\( a = 3 \)[/tex] and [tex]\( b \neq 5 \)[/tex].
- The system is consistent and dependent when [tex]\( a = 3 \)[/tex] and [tex]\( b = 5 \)[/tex].
