Solve by the addition method. (If there is no solution, enter NO SOLUTION. Use the parameters x and y as necessary.)
x − 2y	 = 	−3
−2x + 4y	 = 	6


Let's solve the system of equations using the **addition method**. The two equations are:

1. \( x - 2y = -3 \)
2. \( -2x + 4y = 6 \)

### Step 1: Add the equations directly.

Notice that if we add the two equations as they are, we can eliminate \(x\) because the coefficients of \(x\) in the two equations are opposites:

\[
(x - 2y) + (-2x + 4y) = -3 + 6
\]

Simplifying:

\[
x - 2x - 2y + 4y = 3
\]

This reduces to:

\[
- x + 2y = 3
\]

### Step 2: Solve for \( x \).

The simplified equation is:

\[
- x + 2y = 3
\]

We can solve for \(x\) by adding \(2y\) to both sides:

\[
x = 2y - 3
\]

### Step 3: Substitute \(x = 2y - 3\) into one of the original equations.

Now substitute \(x = 2y - 3\) into the first equation, \( x - 2y = -3 \):

\[
(2y - 3) - 2y = -3
\]

Simplifying:

\[
2y - 3 - 2y = -3
\]

This simplifies to:

\[
-3 = -3
\]

### Step 4: Interpret the result.

Since this is a true statement, it means that the system has infinitely many solutions. The equations are dependent (they represent the same line), and every point on that line is a solution.

### Final Answer: Infinitely many solutions.

The solution is a **parametric solution** where \(x = 2y - 3\), with \(y\) as a free parameter.

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Would you like more details or have any questions?

Here are some related problems for further practice:

1. Solve the system using substitution: \(x + y = 4\) and \(2x - y = 1\).
2. Solve the system using elimination: \(3x + 2y = 6\) and \(2x - 2y = 4\).
3. Check if the system \(x + y = 5\) and \(2x + 2y = 10\) has infinitely many solutions.
4. Solve \(x - y = 7\) and \(x + 2y = 3\) using the addition method.
5. Find the intersection point of \(x + 3y = 9\) and \(2x - y = 4\).

**Tip:** If you obtain a true statement like \(0 = 0\) or \(-3 = -3\) after eliminating variables, it typically means the system has infinitely many solutions.

Solve by the addition method. (If there is no solution, enter NO SOLUTION. Use the parameters x and y as necessary.) x − 2y = −3, −2x + 4y = 6

Learn how to solve the system of linear equations x − 2y = −3 and −2x + 4y = 6 using the addition method. This step-by-step guide covers solving for x and y and explains why the system has infinitely many solutions, making it ideal for Grades 9-11 students.

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