To solve the system of equations using the Elimination method, we need to eliminate one of the variables by adding or subtracting the equations. Here are the given equations:
1. [tex]\(2x - 3y = -4\)[/tex]
2. [tex]\(-4x + 6y = 1\)[/tex]
### Step 1: Align the equations
The equations are already aligned in terms of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[
\begin{aligned}
2x - 3y &= -4 \quad \text{(Equation 1)} \\
-4x + 6y &= 1 \quad \text{(Equation 2)}
\end{aligned}
\][/tex]
### Step 2: Make the coefficients of [tex]\(x\)[/tex] or [tex]\(y\)[/tex] equal
We notice that the coefficients of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in the equations are multiples of each other. Specifically, the coefficients of [tex]\(x\)[/tex] in Equation 2 is [tex]\(-2\)[/tex] times that in Equation 1. We can use this relationship to eliminate [tex]\(x\)[/tex].
### Step 3: Eliminate [tex]\(x\)[/tex]
To eliminate [tex]\(x\)[/tex], we can add Equation 2 to Equation 1 after multiplying Equation 1 by 2:
Multiply Equation 1 by 2:
[tex]\[
2(2x - 3y) = 2(-4) \\
4x - 6y = -8 \quad \text{(Equation 3)}
\][/tex]
So now we have:
[tex]\[
\begin{aligned}
4x - 6y &= -8 \quad \text{(Equation 3)} \\
-4x + 6y &= 1 \quad \text{(Equation 2)}
\end{aligned}
\][/tex]
Add Equation 3 and Equation 2 together:
[tex]\[
(4x - 6y) + (-4x + 6y) = -8 + 1 \\
4x - 4x - 6y + 6y = -7 \\
0 = -7
\][/tex]
### Step 4: Determine the nature of the solution
The resulting equation is [tex]\(0 = -7\)[/tex], which is a contradiction. This means that there is no set of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that can satisfy both equations simultaneously.
### Conclusion
The system of equations is inconsistent, and therefore, there is no solution.
The final answer is:
- No Solution
