Solve the System: Fractions and Integers in x/2 + y = 3 and x + 2y = 6

$\frac{x}{2}+y=3$

$x+2y=6$

To solve this problem, we'll follow these steps:

• Step 1: Solve the first equation for $x$.
• Step 2: Substitute this expression for $x$ into the second equation.
• Step 3: Analyze the resulting equation for $y$.
• Step 4: Determine the relationship between the two equations.

Now, let's work through each step:
Step 1: Starting with the first equation $\frac{x}{2} + y = 3$, we solve for $x$ by isolating it:
Subtract $y$ from both sides:
$\frac{x}{2} = 3 - y$
Multiply both sides by 2 to solve for $x$:
$x = 2(3 - y) = 6 - 2y$

Step 2: Substitute $x = 6 - 2y$ into the second equation $x + 2y = 6$:
$(6 - 2y) + 2y = 6$
Simplify:
$6 = 6$

Step 3: The equation $6 = 6$ is always true, indicating there is no contradiction and hence infinitely many solutions when both conditions arise from manipulating consistent equations.

Step 4: Since manipulating these equations leads us to an identity, they are dependent; both equations are forms of the same linear equation $\frac{x}{2} + y = 3$. Each point on this line satisfies both equations, confirming infinite solutions.

Therefore, the solution to the system of equations is infinite solutions.