How many solutions does the following system have?
{ 16x - 8y=20
{ 4x - 2y= 5
A. 0
B. 1
C. 2
D. Infinitely many
Did you notice that if you multiply the 2nd equation by 4, you get the second?
So , you basically only have one equation, thus
an infinite number of solutions.
To determine the number of solutions for this system of equations, we can use the method of elimination or substitution.
Let's start by applying the method of elimination to this system:
Step 1: Multiply the second equation by 4 to make the coefficients of the x terms the same in both equations:
16x - 8y = 20
16x - 8y = 20
Step 2: Subtract the equations to eliminate the x terms:
(16x - 8y) - (16x - 8y) = 20 - 20
0 = 0
Step 3: Simplify the equation. Since we get an identity (0 = 0), this means all the variables have canceled out, and the system is dependent.
A dependent system indicates that the two equations represent the same line. Thus, there are infinitely many solutions for this system.
Therefore, the correct answer is:
D. Infinitely many