Question

For the given system of equations, what must the values of 'a' and 'c' be given that the system has an infinite number of solutions?

5x - 8y = 12a
x + 8y = c

Option 1: a = -2, c = -4
Option 2: a = -2, c = 4
Option 3: a = 2, c = -4
Option 4: a = 2, c = 4

Answer 1

To have an infinite number of solutions for the given system of equations, 'a' must be 2 and 'c' must be 4.

Step-by-step explanation:

To have an infinite number of solutions for the given system of equations, the two equations must be equivalent, meaning they represent the same line on the coordinate plane. In order for this to happen, the coefficients of x and y in both equations must be proportional. Let's compare the two equations:

5x - 8y = 12a

x + 8y = c

The coefficients of x in both equations are not proportional, so we need to make them proportional by finding a common factor that can be multiplied to both equations. The common factor is 8. Multiply the second equation by 8:

8(x + 8y) = 8c

8x + 64y = 8c

Now compare this equation to the first equation:

5x - 8y = 12a

8x + 64y = 8c

By comparing the coefficients of x and y, we can see that 5x and 8x are proportional, and -8y and 64y are also proportional. Therefore, 'a' must be 2 and 'c' must be 4 for the system of equations to have an infinite number of solutions.