2. Beckett needs to solve the system of equations below using elimination. \[ -2 x+4 y=-2 \] \[ 6 x-y=28 \] Which correctly describes the first step Beckett should take? A. Multiply each term in the \( 1^{\text {st }} \) equation by -3 B. Multiply each term in the \( 1^{\text {st }} \) equation by 3 C. Multiply each term in the \( 2^{\text {nd }} \) equation by -4 D. Both B and C would work

To solve the system of equations using the elimination method, Beckett needs to manipulate the equations so that one of the variables can be eliminated by addition or subtraction. Let's analyze the given system: \[ -2x + 4y = -2 \quad \text{(Equation 1)} \] \[ 6x - y = 28 \quad \text{(Equation 2)} \] **Objective:** Eliminate one of the variables (either \( x \) or \( y \)) to solve for the other. **Analyzing the Options:** 1. **Option A: Multiply each term in the \( 1^{\text{st}} \) equation by -3** - This would transform Equation 1 to: \[ 6x - 12y = 6 \] - While this is a valid transformation, it's not paired with another step in the options provided. 2. **Option B: Multiply each term in the \( 1^{\text{st}} \) equation by 3** - This transforms Equation 1 to: \[ -6x + 12y = -6 \] - Adding this to Equation 2 (\( 6x - y = 28 \)) results in: \[ 11y = 22 \quad \Rightarrow \quad y = 2 \] 3. **Option C: Multiply each term in the \( 2^{\text{nd}} \) equation by -4** - This transforms Equation 2 to: \[ -24x + 4y = -112 \] - Adding this to Equation 1 (\( -2x + 4y = -2 \)) results in: \[ -26x + 8y = -114 \quad \Rightarrow \quad \text{(Further simplification needed)} \] - Alternatively, subtracting Equation 1 from the transformed Equation 2 eliminates \( y \): \[ -24x + 4y - (-2x + 4y) = -112 - (-2) \quad \Rightarrow \quad -22x = -110 \quad \Rightarrow \quad x = 5 \] **Conclusion:** Both **Option B** and **Option C** are valid first steps in the elimination process. Multiplying the first equation by 3 allows for the elimination of \( x \) when added to the second equation, and multiplying the second equation by -4 allows for the elimination of \( x \) when added to the first equation. Therefore, both approaches are correct. **Answer:** **D. Both B and C would work**