The sum of two numbers is 95 . If the larger number is increased by twice the smaller number, the result is 120 . What is the larger number? If \( S \) = the smaller number and \( L \) = the larger number, then which of the following systems of equations represents the word problem? \( S+L=95 \) and \( S+2 L=120 \) \( S+L=95 \) and \( 2(S+L)=120 \) \( S+L=95 \) and \( 2 S+L=120 \)

To solve the problem, we need to extract the known conditions and set up the correct system of equations based on the information provided. 1. **Extracting Known Conditions:** - The sum of two numbers (smaller number \( S \) and larger number \( L \)) is 95: \[ S + L = 95 \] - If the larger number \( L \) is increased by twice the smaller number \( S \), the result is 120: \[ L + 2S = 120 \] 2. **Identifying the Correct System of Equations:** - The first equation is \( S + L = 95 \). - The second equation is \( L + 2S = 120 \). Thus, the correct system of equations that represents the word problem is: \[ S + L = 95 \quad \text{and} \quad L + 2S = 120 \] Now, let's solve this system of equations to find the larger number \( L \). ### Step-by-Step Solution: 1. From the first equation, we can express \( L \) in terms of \( S \): \[ L = 95 - S \] 2. Substitute \( L \) into the second equation: \[ (95 - S) + 2S = 120 \] 3. Simplify the equation: \[ 95 - S + 2S = 120 \] \[ 95 + S = 120 \] 4. Solve for \( S \): \[ S = 120 - 95 \] \[ S = 25 \] 5. Now substitute \( S \) back into the equation for \( L \): \[ L = 95 - S = 95 - 25 = 70 \] ### Final Answer: The larger number \( L \) is \( 70 \). Thus, the correct system of equations is: - \( S + L = 95 \) - \( L + 2S = 120 \)