Judy has \( \$ 20 \) and is saving at a rate of \( \$ 6 \) per week. Ida has \( \$ 172 \) and is spending at a rate of \( \$ 4 \) per week. After how many weeks will each have the same amount of money? Write an equation using \( x \) and \( y \) for Judy and Ida. What does \( x \) represent? What does \( y \) represent? bolve this problem using any method you choose.

Let's define Judy's and Ida's funds over time. Judy starts with \( \$20 \) and saves \( \$6 \) per week, leading to the equation: \[ y = 20 + 6x \] Ida starts with \( \$172 \) and spends \( \$4 \) per week, described by the equation: \[ y = 172 - 4x \] In these equations, \( x \) represents the number of weeks, and \( y \) represents the amount of money in dollars that either Judy or Ida has after those weeks. To find out when they will have the same amount, we can set the two equations equal: \[ 20 + 6x = 172 - 4x \] Solving for \( x \): 1. Add \( 4x \) to both sides: \[ 20 + 10x = 172 \] 2. Subtract \( 20 \) from both sides: \[ 10x = 152 \] 3. Divide both sides by \( 10 \): \[ x = 15.2 \] Since we generally deal with whole weeks, we can interpret \( x = 15 \) as the last complete week where their amounts won't match, and \( x = 16 \) as the week where they'll equalize. Now, plugging \( x = 16 \) back into either equation to find \( y \): For Judy: \[ y = 20 + 6(16) = 20 + 96 = 116 \] For Ida: \[ y = 172 - 4(16) = 172 - 64 = 108 \] So after 16 weeks, Judy will have \( \$116 \), and Ida will have \( \$108\), meaning they won’t have the same amounts at whole weeks. However, they will meet at \( $116 \) soon after 15 weeks' end when we factor in partial weeks. Now, the question beckons how many weeks until they meet: Technically, it means they meet halfway at about \(16\) weeks based on \(15.2\).