\( J o h n \) is twice as old as Mary. The sum of their ages is 52 . How old is Mary? If \( J= \) John's age and \( M= \) Mary's age, which system of equations could be used to solve the problem? \[ J=M \text { and } J+M=52 \] \( J=2 M \) and \( J M=52 \) \( J \) and \( J+M=52 \)

To solve the problem, we need to extract the known conditions from the information given: 1. John is twice as old as Mary: This can be expressed as the equation \( J = 2M \). 2. The sum of their ages is 52: This can be expressed as the equation \( J + M = 52 \). Now, let's identify the correct system of equations from the options provided: 1. \( J = M \) and \( J + M = 52 \) - This is incorrect because it states that John and Mary are the same age. 2. \( J = 2M \) and \( J M = 52 \) - This is incorrect because the second equation should be a sum, not a product. 3. \( J \) and \( J + M = 52 \) - This is incomplete and does not provide a second equation. The correct system of equations that can be used to solve the problem is: - \( J = 2M \) - \( J + M = 52 \) Now, let's solve this system step by step. 1. Substitute \( J \) from the first equation into the second equation: \[ 2M + M = 52 \] 2. Combine like terms: \[ 3M = 52 \] 3. Solve for \( M \): \[ M = \frac{52}{3} \] Now, let's calculate \( M \) to find Mary's age. Calculate the value by following steps: - step0: Calculate: \(\frac{52}{3}\) Calculating \( M \) gives us: \[ M = \frac{52}{3} \approx 17.33 \] This means Mary is approximately \( 17 \frac{1}{3} \) years old, or about 17 years and 4 months. To summarize: - The correct system of equations is \( J = 2M \) and \( J + M = 52 \). - Mary's age is approximately \( 17.33 \) years.