\( \begin{array}{l}5 x+2 y=8 \\ x+y=4 \\ \text { If you want to solve the system of equations by addition, which of the following could you do? } \\ \text { Multiply the second equation by }-2 \text { and add. } \\ \text { Multiply the second equation by } 5 \text { and add. } \\ \text { Multiply the second equation by } 2 \text { and add. }\end{array} . \begin{array}{l}\end{array} \).

To solve the system of equations using the method of addition (also known as the elimination method), we need to manipulate the equations so that adding them will eliminate one of the variables. The given system of equations is: 1. \( 5x + 2y = 8 \) (Equation 1) 2. \( x + y = 4 \) (Equation 2) ### Step 1: Analyze the options We will consider each option provided to see if it will allow us to eliminate one of the variables when we add the equations. 1. **Multiply the second equation by \(-2\)**: \[ -2(x + y) = -2(4) \implies -2x - 2y = -8 \] Now, we can add this to the first equation: \[ (5x + 2y) + (-2x - 2y) = 8 - 8 \] This simplifies to: \[ 3x = 0 \implies x = 0 \] This option works. 2. **Multiply the second equation by \(5\)**: \[ 5(x + y) = 5(4) \implies 5x + 5y = 20 \] Now, we can add this to the first equation: \[ (5x + 2y) + (5x + 5y) = 8 + 20 \] This simplifies to: \[ 10x + 7y = 28 \] This option does not eliminate a variable. 3. **Multiply the second equation by \(2\)**: \[ 2(x + y) = 2(4) \implies 2x + 2y = 8 \] Now, we can add this to the first equation: \[ (5x + 2y) + (2x + 2y) = 8 + 8 \] This simplifies to: \[ 7x + 4y = 16 \] This option does not eliminate a variable. ### Conclusion The only option that allows us to eliminate a variable when we add the equations is: - **Multiply the second equation by \(-2\) and add.** Thus, the correct answer is: **Multiply the second equation by \(-2\) and add.**