Understanding the Problem
To determine the values of a and b for which the equations have infinitely many solutions, we start with the given equations:
1. 2x - 3y = 7
2. (a + b)x - (a + b - 3)y = 4a + b
Condition for Infinitely Many Solutions
For a system of linear equations to have infinitely many solutions, the two equations must be dependent. This means that one equation can be expressed as a multiple of the other.
Setting Up the Ratios
We can express the two equations in the standard form:
- First equation: 2x - 3y = 7
- Second equation: (a + b)x - (a + b - 3)y = 4a + b
To find the dependency, we set up the ratios of coefficients:
- Coefficients of x and y must be equal, and their constants must also satisfy the same ratio.
Establishing the Ratios
1. Coefficients of x: (a + b) / 2 = (a + b - 3) / -3
2. Constants: (4a + b) / 7
Setting the first ratio equal to the second:
(a + b) / 2 = (4a + b) / 7
Cross Multiplying
Cross multiply to eliminate the fractions:
7(a + b) = 2(4a + b)
Expanding gives:
7a + 7b = 8a + 2b
Solving for a and b
Rearranging terms:
7b - 2b = 8a - 7a
5b = a
Conclusion
The relationship between a and b for infinitely many solutions is:
a = 5b
Choose any value for b, then a can be calculated as 5 times that value. For example, if b = 1, then a = 5. Thus, there are infinitely many solutions based on this relationship.