We are given the system of linear equations:
[tex]\[ y = 4.5x + 3 \][/tex]
[tex]\[ y = C(x + 2) \][/tex]
To determine the values of [tex]\( C \)[/tex] for which the system has no solution, we need to analyze the conditions under which the lines represented by these equations are parallel but not coincident (i.e., they have the same slope but different y-intercepts).
### Step-by-Step Solution:
1. Identify the Slopes:
- The slope of the first equation [tex]\( y = 4.5x + 3 \)[/tex] is [tex]\( 4.5 \)[/tex].
- The second equation can be rewritten in slope-intercept form:
[tex]\[ y = Cx + 2C \][/tex]
Therefore, the slope of the second equation is [tex]\( C \)[/tex].
2. Set the Slopes Equal:
For the lines to be parallel, their slopes must be equal:
[tex]\[ 4.5 = C \][/tex]
So, the value of [tex]\( C \)[/tex] must be [tex]\( 4.5 \)[/tex].
3. Check the Y-Intercepts:
- The y-intercept of the first equation is [tex]\( 3 \)[/tex].
- The y-intercept of the second equation (when [tex]\( C = 4.5 \)[/tex]) can be calculated as follows:
[tex]\[ y\text{-intercept} = 2C = 2 \times 4.5 = 9 \][/tex]
4. Ensure Different Y-Intercepts:
For the system to have no solution, the y-intercepts must be different. In this scenario:
[tex]\[ 3 \neq 9 \][/tex]
Since the y-intercepts are different, the two lines are parallel but not coincident, meaning they do not intersect. Hence, there is no solution to the system of equations.
### Conclusion:
The system of equations has no solution when [tex]\( C = 4.5 \)[/tex].
