Question

Use the substitution method to solve the system x−2y=3
, 4x−8y=12
.(1 point)
Responses

(3,0)
left parenthesis 3 comma 0 right parenthesis

infinite solutions
infinite solutions

(0,3)
left parenthesis 0 comma 3 right parenthesis

no solution

Answer 1

To solve the system using the substitution method, we'll start with the first equation and express one variable in terms of the other.

The given equations are:
1) \( x - 2y = 3 \)
2) \( 4x - 8y = 12 \)

### Step 1: Solve the first equation for \( x \).
From the first equation:
\[ x = 2y + 3 \]

### Step 2: Substitute into the second equation.
Now substitute \( x \) from the first equation into the second equation:
\[ 4(2y + 3) - 8y = 12 \]

### Step 3: Simplify and solve for \( y \).
Distributing the 4:
\[ 8y + 12 - 8y = 12 \]
This simplifies to:
\[ 12 = 12 \]

This statement is always true, which indicates that the two equations represent the same line. Hence, there are infinite solutions.

### Conclusion:
The system has infinite solutions.

The correct response is:
**infinite solutions**