Question

Ax+by=12

2x+8y=60
in the system of equations above, a and b are constants. if the system has infinitely many solutions, what is the value of a/b?

Answer 1

For the system of equations to have infinitely many solutions, the equations must represent the same line. Upon transforming the second equation, it becomes apparent that 'a' must be 2, and 'b' must be 8, leading to the calculation that a/b is 1/4.

Step-by-step explanation:

The student is working with a system of linear equations with constants a and b. For the system to have infinitely many solutions, both equations must represent the same line. This means that the coefficients of the corresponding variables must be proportional.

The second equation can be transformed by dividing every term by 2, resulting in x + 4y = 30. Comparing this to the first equation ax + by = 12, we can see that for them to be the same line, the coefficients of x and y in both equations must be proportional, and the constants must also align.

Thus, a/1 = b/4 = 12/30. Simplifying the fraction on the right side gives us 2/5. Therefore, equating a/1 to 2/5, we find that a must equal 2. Equating b/4 to 2/5, we find that b must equal 8. As a result, the value of a/b equals 2/8, which simplifies to 1/4.

Answer 2

The value of a/b where the system of equations ax+by=12 and 2x+8y=60 has infinitely many solutions is 0.25, because the two equations represent the same line and the coefficients must be proportional.

Step-by-step explanation:

If the system of equations ax+by=12 and 2x+8y=60 has infinitely many solutions, it means that the two equations are actually the same line, just written in different forms. This implies that the ratios of the coefficients of x and y, as well as the constant terms in both equations, must be equal.

To find the value of a/b, we should transform the second equation into a similar form as the first. Dividing the entire second equation by 2 gives us x+4y=30.

Next, we compare the coefficients: the coefficient of x in the second equation is now 1, which means that a must also be 1. The coefficient of y is 4, so b must be 4. Therefore, the value of a/b is 1/4 or 0.25.