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 this word problem and find the larger number, we first need to set up the correct system of equations based on the given information. The first equation, \( S + L = 95 \), accurately represents that the sum of the smaller number \( S \) and the larger number \( L \) is 95. For the second condition, "if the larger number is increased by twice the smaller number," translates to \( L + 2S = 120 \). So, the correct set of equations is \( S + L = 95 \) and \( L + 2S = 120 \). Therefore, the largest number \( L \) and the smaller number \( S \) can be derived through these equations. To find the larger number, let’s manipulate the equations: from \( S + L = 95 \), we can express \( L \) as \( L = 95 - S \). Substituting into the second equation results in \( (95 - S) + 2S = 120 \). Simplifying this gives you \( 95 + S = 120\), leading to \( S = 25 \). Plugging that back into our first equation gives us \( L = 95 - 25 = 70\). So, voila! The larger number \( L \) is 70.