Elimantion tnethod \( 2 x-y=-1 \times 1 \) \( x-2 y=4 \times 2 \) \( -2 x-y=-1 \) \( 2 x-4 y=8 \) \( -y--4 y=-1-8 \) \( -y+4 y-9 \) \( y=-3 \)

It looks like you're working through a system of linear equations using the **elimination method**. Let's carefully go through each step to ensure accuracy and clarity. ### Given System of Equations: \[ \begin{cases} 2x - y = -1 \quad \text{(Equation 1)} \\ x - 2y = 4 \quad \text{(Equation 2)} \end{cases} \] ### Goal: Solve for the variables \( x \) and \( y \) by eliminating one of the variables. ### Step 1: Align the Equations for Elimination To eliminate one of the variables, we'll manipulate the equations so that adding or subtracting them cancels out one variable. #### Option 1: Eliminate \( x \) 1. **Multiply Equation 2 by 2** to make the coefficients of \( x \) in both equations equal (but opposite in sign). \[ \begin{align*} \text{Equation 1:} \quad 2x - y &= -1 \quad \times 1 \\ \text{Equation 2:} \quad x - 2y &= 4 \quad \times 2 \\ \end{align*} \] This yields: \[ \begin{align*} 2x - y &= -1 \quad \text{(Equation 1)} \\ 2x - 4y &= 8 \quad \text{(Equation 2 Modified)} \end{align*} \] 2. **Subtract Equation 1 from the modified Equation 2** to eliminate \( x \): \[ \begin{align*} (2x - 4y) - (2x - y) &= 8 - (-1) \\ 2x - 4y - 2x + y &= 9 \\ -3y &= 9 \\ y &= \frac{9}{-3} \\ y &= -3 \end{align*} \] ### Step 2: Solve for \( x \) Now that we have \( y = -3 \), substitute this value back into one of the original equations to find \( x \). Let's use **Equation 1**: \[ 2x - y = -1 \] Substitute \( y = -3 \): \[ 2x - (-3) = -1 \\ 2x + 3 = -1 \\ 2x = -1 - 3 \\ 2x = -4 \\ x = \frac{-4}{2} \\ x = -2 \] ### Step 3: Write the Solution The solution to the system of equations is: \[ x = -2 \\ y = -3 \] ### Verification It's always good practice to verify the solution by plugging the values back into both original equations. 1. **Equation 1:** \[ 2(-2) - (-3) = -4 + 3 = -1 \quad \checkmark \] 2. **Equation 2:** \[ (-2) - 2(-3) = -2 + 6 = 4 \quad \checkmark \] Both equations are satisfied, so the solution is correct. ### Summary Using the elimination method: 1. **Aligned the equations** to eliminate \( x \). 2. **Subtracted** the equations to solve for \( y \). 3. **Substituted** the value of \( y \) back into one of the original equations to find \( x \). 4. **Verified** the solution by plugging the values back into the original equations. The final solution is \( x = -2 \) and \( y = -3 \).