Question

Given the pair of equations, select the solution:

4x+8y=12

2x+2y=6

a
(0,3)

b
(3,0)

c
no solution

d
infinate solution

Answer 1

To determine the solution of the given pair of equations:

1. **Equations**:
- \(4x + 8y = 12\) (Equation 1)
- \(2x + 2y = 6\) (Equation 2)

2. **Simplifying**:
Let's simplify both equations.

- For Equation 1, divide everything by 4:
\[
x + 2y = 3 \quad \text{(Equation 1 simplified)}
\]

- For Equation 2, divide everything by 2:
\[
x + y = 3 \quad \text{(Equation 2 simplified)}
\]

3. **Express y in terms of x**:
- From the simplified Equation 1:
\[
y = \frac{3 - x}{2} \quad \text{(From Equation 1)}
\]

- From the simplified Equation 2:
\[
y = 3 - x \quad \text{(From Equation 2)}
\]

4. **Setting them equal to find corresponding solutions**:
We now set the two expressions for \(y\) equal to each other:
\[
\frac{3 - x}{2} = 3 - x
\]

To eliminate the fraction, multiply through by 2:
\[
3 - x = 2(3 - x)
\]
This expands to:
\[
3 - x = 6 - 2x
\]

Rearranging gives us:
\[
x - 2x = 6 - 3
\]
\[
-x = 3
\]
\[
x = -3
\]

5. **Finding y**:
Substitute \(x = -3\) into either equation to find \(y\):
Using the simplified Equation 2:
\[
y = 3 - (-3) = 3 + 3 = 6
\]
So the solution is: \((-3, 6)\).

6. **Checking for equivalence of equations**:
Notice that if we simplify:
- \(2x + 2y = 6\) (multiply by 2) gives \(4x + 4y = 12\), which when compared with \(4x + 8y = 12\), indicates the two are actually inconsistent since \(8y\) cannot equal \(4y\) under usual real numbers.

7. **Conclusion**:
The original equations do not represent parallel lines since they can be coinciding or lead to a unique solution. Thus, they current equivalently represent a unique point or an intersection if summed correctly.

However, here we directly see they are conflicting indicating the equations align parallel without solving at unique intersection.

Therefore, the correct conclusion is:
**Answer**: c) no solution.