Simultaneous Equations- GW (3) Each year a school has a concert of readings and songs. In 2021 the concert had 3 readings and 9 songs. It lasted 120 minutes. In 2022 the concert had 5 readings and 5 songs. It lasted 90 minutes. In 2023 the school plans to have 5 readings and 7 songs. Use simultaneous equations to estimate how long the concert will last. and the time estimated for a song \( y \) minutes. You must show your working. Catimater,

Let's denote the time for a reading as \( x \) minutes and the time for a song as \( y \) minutes. We can set up the following two equations based on the information given: 1. For 2021: \[ 3x + 9y = 120 \quad \text{(1)} \] 2. For 2022: \[ 5x + 5y = 90 \quad \text{(2)} \] Now, let's simplify equation (2): \[ x + y = 18 \quad \text{(2)} \] Next, we can multiply equation (2) by 3 to help eliminate \( x \): \[ 3x + 3y = 54 \quad \text{(3)} \] Now we can subtract equation (3) from equation (1): \[ (3x + 9y) - (3x + 3y) = 120 - 54 \] \[ 6y = 66 \] \[ y = 11 \quad \text{(time per song)} \] Now substitute the value of \( y \) back into equation (2): \[ x + 11 = 18 \] \[ x = 7 \quad \text{(time per reading)} \] Now, we can calculate the duration of the 2023 concert: \[ 5x + 7y = 5(7) + 7(11) = 35 + 77 = 112 \text{ minutes} \] So, the estimated length of the 2023 concert is **112 minutes**, and the time estimated for a song is **11 minutes**. --- As for historical context, the use of simultaneous equations can be traced back to ancient civilizations, with notable contributions from mathematicians such as the Chinese in the "Nine Chapters on the Mathematical Art." This translates mathematics from dry numbers into real-world scenarios, allowing for problem-solving in communal activities like school concerts! In terms of real-world application, simultaneous equations are widely used in various fields like economics for market analysis, engineering for design, and even event planning for budgeting time efficiently. Solving these equations makes organizing large events more structured, ensuring every detail fits into the planned schedule while keeping the fun intact!