Problem 235 - Elementary Algebra

Question

At a school concert, 425 tickets were sold. Student tickets cost \(\$ 5\) each and adult tickets cost \(\$ 8\) each. The total receipts for the concert were \(\$ 2,851\). Solve the system \(\left\\{\begin{array}{l}s+a=425 \\ 5 s+8 a=2,851\end{array}\right.\) to find \(s,\) the number of student tickets and \(a,\) the number of adult tickets.

Step-by-Step Solution

Answer

183 student tickets, 242 adult tickets

Step 1: Define Variables

Let T represent the number of student tickets and A represent the number of adult tickets.

Step 2: Write Equations

Based on the problem, write two equations: 1. The total number of tickets sold is 425: \[ T + A = 425 \]2. The total receipts from the tickets are 2,851: \[ 5T + 8A = 2851 \]

Step 3: Solve the First Equation for One Variable

Solve the first equation for T: \[ T = 425 - A \]

Step 4: Substitute into the Second Equation

Substitute the expression for T from the first equation into the second equation: \[ 5(425 - A) + 8A = 2851 \]

Step 5: Simplify the Equation

Expand and simplify the equation: \[ 2125 - 5A + 8A = 2851 \]Combine like terms: \[ 2125 + 3A = 2851 \]

Step 6: Solve for A

Isolate A by first subtracting 2125 from both sides: \[ 3A = 2851 - 2125 \]\[ 3A = 726 \]Next, divide both sides by 3: \[ A = \frac{726}{3} = 242 \]

Step 7: Find T

Substitute A back into the equation T = 425 - A: \[ T = 425 - 242 \]\[ T = 183 \]

Step 8: Verify the Solution

Verify that the solution satisfies both original equations:1. \[ 183 + 242 = 425 \] True2. \[ 5(183) + 8(242) = 2851 \] TrueBoth are true, so the solution is correct.

Key Concepts

linear equations

Linear equations are equations of the first degree, meaning they involve no exponents higher than one. They typically have the form (ax + by = c), where a, b, and c are constants. In this exercise, we had two linear equations derived from the problem scenario: the total count of tickets sold and the total sales receipts. The equations are (T + A = 425) and (5T + 8A = 2851), where T is the number of student tickets and A is the number of adult tickets.

substitution method

The substitution method is a technique for solving a system of linear equations. It involves solving one equation for one variable and then substituting that expression into the other equation. In our case, we first solved (T + A = 425) for T, getting (T = 425 - A). We then replaced T with (425 - A) in the second equation (5T + 8A = 2851). This allowed us to solve the second equation using one variable, simplifying our work.

algebraic problem solving

Algebraic problem-solving often involves translating word problems into mathematical equations. For our ticket sales scenario, we translated the information into two linear equations: the total number of tickets and the total sales amount. We used algebraic techniques like solving for variables and substitution to find the number of student and adult tickets required. This method of breaking down the problem into manageable pieces is key in many algebraic solutions.

ticket sales problem

Ticket sales problems are common in algebra as they involve real-life scenarios with quantities and costs. Here, student tickets and adult tickets form our key variables. Knowing the total tickets sold (425) and the total amount collected (2851 dollars) gave us enough information to set up our linear equations. Solving these equations helped us determine the number of each type of ticket sold. Such problems help us understand how to extract mathematical equations from everyday situations.

receipts calculation

Calculating receipts involves summing up the total money collected from sales. In this context, we expressed the total receipts as (5T + 8A = 2851), where 5T is the money from student tickets and 8A is the money from adult tickets. With our equations set, we used the defined prices to break down the total receipts, helping us identify the number of student and adult tickets sold. Receipts calculations like this show how mathematical equations can help manage and analyze financial data.