Part A: To determine the relationship between 12 and 6, we can compare the given numbers. In this case, since both 12 and 6 are equal, we can say that the relationship between them is that they are equal (12 = 6).
Part B: To determine if the two equations have one solution, no solution, or infinitely many solutions, we need to solve the system of equations. Let's solve them algebraically:
Equation 1: 4x + 6y = 12
Equation 2: 2x + 3y = 6
We can solve this system of equations using the method of substitution or elimination.
Using the method of elimination:
Multiply Equation 2 by 2 to make the coefficients of x the same.
Equation 2 becomes:
4x + 6y = 12
Now, we have two equations with the same coefficients of x:
4x + 6y = 12 (Equation 1)
4x + 6y = 12 (Equation 2)
Since both equations are identical, we can say that the system of equations has infinitely many solutions.
Part C: We can verify our answers to Part A and B by solving algebraically. From our previous solution, we determined that the system has infinitely many solutions. To verify this, we can choose any pair of values for x and y that satisfy the original equations.
Let's choose x = 0 and y = 2:
Substituting these values into the original equations:
Equation 1: 4(0) + 6(2) = 12
Simplifying, we get: 0 + 12 = 12, which is true.
Equation 2: 2(0) + 3(2) = 6
Simplifying, we get: 0 + 6 = 6, which is also true.
Since both equations are true, our chosen values of x = 0 and y = 2 satisfy both equations, confirming that the system has infinitely many solutions.