solve this simutaneous equation using the elimination method

To solve the given system of simultaneous equations using the elimination method, we'll follow these steps:

### Given System:
\[
\begin{aligned}
1. \quad & 3x = y - 1 \quad \text{(Equation 1)} \\
2. \quad & 2y - 2x = 3 \quad \text{(Equation 2)}
\end{aligned}
\]

### Step 1: Rearrange both equations in standard form
Let's rearrange both equations so they are in the form \(ax + by = c\):

**Equation 1:**
\[
3x - y = -1
\]

**Equation 2:**
\[
-2x + 2y = 3
\]

### Step 2: Make the coefficients of \( y \) or \( x \) the same
To eliminate one of the variables, we want the coefficients of either \( x \) or \( y \) to be the same (with opposite signs) so that when we add or subtract the equations, one variable will cancel out.

Let's eliminate \( y \). To do this, multiply Equation 1 by 2 to match the coefficient of \( y \) in Equation 2:

\[
2(3x - y) = 2(-1)
\]

This gives us:

\[
6x - 2y = -2 \quad \text{(Equation 3)}
\]

Now, the system of equations looks like this:

\[
\begin{aligned}
6x - 2y & = -2 \quad \text{(Equation 3)} \\
-2x + 2y & = 3 \quad \text{(Equation 2)}
\end{aligned}
\]

### Step 3: Add the equations to eliminate \( y \)
Now, add Equation 3 and Equation 2 to eliminate \( y \):

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

Simplify:

\[
4x = 1
\]

### Step 4: Solve for \( x \)
\[
x = \frac{1}{4}
\]

### Step 5: Substitute \( x \) back into one of the original equations
Substitute \( x = \frac{1}{4} \) into Equation 1 to solve for \( y \):

\[
3\left(\frac{1}{4}\right) - y = -1
\]

This simplifies to:

\[
\frac{3}{4} - y = -1
\]

Subtract \( \frac{3}{4} \) from both sides:

\[
-y = -1 - \frac{3}{4} = -\frac{7}{4}
\]

Multiply by -1:

\[
y = \frac{7}{4}
\]

### Final Answer:
The solution to the system is:

\[
x = \frac{1}{4}, \quad y = \frac{7}{4}
\]

Would you like any further explanations or details on the elimination method?

### Related Questions:
1. How would the solution change if we used the substitution method instead?
2. What is the geometric interpretation of the solution to this system?
3. How can we verify the solution using a graphical approach?
4. What is the impact of multiplying the entire equation by a scalar during elimination?
5. How can elimination be used to solve systems with three variables?

### Tip:
When using the elimination method, always aim to eliminate one variable by making the coefficients equal in magnitude but opposite in sign. This simplifies the process and helps to avoid mistakes.

Learn how to solve a system of simultaneous equations using the elimination method with a step-by-step solution. This problem covers key algebraic concepts, including simultaneous equations and the elimination method. Suitable for high school students.

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