What is the solution to the system of equations 2x - y = 1 and 4x - 2y = 2?

To solve the system of equations:

\[
\begin{align*}
2x - y & = 1 \quad \text{(Equation 1)} \\
4x - 2y & = 2 \quad \text{(Equation 2)}
\end{align*}
\]

We can solve this system using either the substitution method or the elimination method. However, let's first analyze the system to determine the best approach.

### Step 1: Analyze the system
Notice that Equation 2 is a multiple of Equation 1:

\[
4x - 2y = 2 \quad \text{(Equation 2)}
\]
can be rewritten as:
\[
2(2x - y) = 2
\]

This implies that Equation 2 is equivalent to Equation 1, just multiplied by 2. Therefore, both equations represent the same line.

### Step 2: Interpret the result
Since the two equations are equivalent, the system does not have a unique solution. Instead, it has infinitely many solutions, meaning every point on the line \( 2x - y = 1 \) is a solution.

### Step 3: Express the solution
The general solution can be written as:

\[
\boxed{y = 2x - 1}
\]

For any value of \( x \), \( y \) will be determined by \( y = 2x - 1 \). This means the solution is a line, not a single point.

### Final Answer:
The system has infinitely many solutions, which can be expressed as:

\[
\boxed{(x, y) = (x, 2x - 1)}
\]

for any real number \( x \).

Would you like to explore this further or have any questions?

---

Here are 5 follow-up questions you might consider:
1. How can you verify that the two equations represent the same line?
2. What happens if you change one of the constants in the equations?
3. How would you represent the solution graphically?
4. What would the solution be if the equations were slightly different, such as \( 2x - y = 1 \) and \( 4x - 2y = 3 \)?
5. How can you recognize when a system of equations has no solution, one solution, or infinitely many solutions?

**Tip:** When dealing with a system of linear equations, check if one equation is a multiple of the other. If they are, the system represents the same line and has infinitely many solutions.

Explore the solution to the system of equations 2x - y = 1 and 4x - 2y = 2 with detailed steps. Learn how to determine infinite solutions in a linear system. Suitable for Grades 9-12 mathematics education.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation