The system
3x + ay = 0
(a-2)x + 5y = 0
has infinitely many solutions.
What is the smallest possible value of a?
Thanks!
To determine the value of a, we need to find the conditions under which the system will have infinitely many solutions. In other words, we need to find the value of a that makes the two equations dependent.
Let's start by writing the equations in standard form:
3x + ay = 0 ...(1)
(a - 2)x + 5y = 0 ...(2)
For the system to have infinitely many solutions, the two equations must be linearly dependent, meaning one equation is a multiple of the other. We can check this by examining the coefficients of x and y.
In equation (1), the coefficient of x is 3, and in equation (2), the coefficient of x is (a - 2). So, if these two coefficients are proportional, meaning they are equal or one is a multiple of the other, the system will have infinitely many solutions.
Setting up the proportion:
3 = k(a - 2)
where k is some constant.
Simplifying the equation:
3 = ka - 2k
We can solve this equation for a.
Rearranging the terms:
ka - 2k - 3 = 0
This is now a linear equation in a. To find the value of a that satisfies this equation, we can solve it using any method we prefer.
For simplicity, let's solve it by factoring:
k(a - 3) - 1(a - 3) = 0
Factoring out the common factor (a - 3):
(a - 3)(k - 1) = 0
Now, we have two possibilities:
1) (a - 3) = 0 --> a = 3
2) (k - 1) = 0 --> k = 1
Since we want to find the smallest possible value of a, we choose the value of a from the first case, which is a = 3.
Therefore, the smallest possible value of a for the system to have infinitely many solutions is a = 3.