Question

Use the following information. Your school drama club is putting on a play next month. By selling tickets for the play, the club hopes to raise \(\$ 600\) for the drama fund for new costumes, scripts, and scenery for future plays. Let \(x\) represent the number of adult tickets they sell at \(\$ 8\) each, and let \(y\) represent the number of student tickets they sell at \(\$ 5\) each. What is the \(y\) -intercept? What does it represent in this situation?

Short answer

The y-intercept is 120. It represents the scenario where if no adult tickets are sold, the number of student tickets that need to be sold to collect $600 for future plays is 120.

Step-by-step solution

Set up the equation

First, lay out the given equation which represents the total amount of money. The equation for the problem is \(8x + 5y = 600\). Here, x represents the number of adult tickets and y the number of student tickets sold.

Calculate the y-intercept

The y-intercept is the value of y when x, the number of adult tickets sold, equals zero. So, in order to find the y-intercept, set x equals zero in the equation: when x=0, \(8(0) + 5y = 600\), which simplifies to: \(5y = 600\). Solve for y by dividing both sides of the equation by 5: \(y = 600 / 5 = 120\). This means that the y-intercept is 120.

Interpret the y-intercept

The y-intercept (120) is the number of student tickets that need to be sold if no adult tickets are sold to raise the $600 needed for future plays. This equates to the club needing to sell 120 student tickets at $5 each to reach the goal.

Key concepts

Y-Intercept

The y-intercept is a fundamental element in the world of linear equations, specifically when graphing a line on the Cartesian plane. It represents the point at which the line crosses the y-axis. To find the y-intercept from an equation, you simply set the value of all other variables to zero and solve for y.

In the context of the drama club's ticket sales, the y-intercept is calculated by setting the number of adult tickets, or x, to zero. The resulting equation, 5y = 600, highlights that the club needs to sell 120 student tickets to achieve its financial goal without selling any adult tickets. This number isn’t arbitrary but gives a clear base value from which the club can strategize their sales. What if they don't sell a single adult ticket? The y-intercept gives reassurance that there's a fallback plan—exactly 120 student tickets will still save the day.

System of Equations

A system of equations is a set of two or more equations with the same variables. The goal usually is to find a solution that satisfies all equations simultaneously. In practical scenarios, these systems can model real-world situations involving constraints and multiple objectives.

Looking at our example, if we had another condition, such as a limited number of tickets, we would introduce a second equation to represent this limit. Together with the original equation 8x + 5y = 600, the second one forms a system that must be solved collectively. Methods such as substitution, elimination, or graphing can be used to find a solution set that represents the number of adult and student tickets that meet all requirements. By mastering systems of equations, you gain the power to tackle complex problems that reflect a multi-faceted reality.

Algebraic Representation

The term algebraic representation refers to using algebraic expressions and equations to symbolically describe relationships among quantities. It is the language of mathematics that allows us to model and solve real-world problems succinctly and systematically.

In the drama club's case, the relationship between ticket sales and revenue generation is expressed as 8x + 5y = 600. This algebraic representation discerns how different ticket sales combinations can meet the club’s financial goal. Each term in this equation carries a significance: 8x stands for revenue from adult tickets, 5y from student tickets, and 600 is the goal. Learning how to sculpt such representations is a vital skill, enabling students to navigate through various challenges by forming clear, workable equations. It's the difference between frantically guessing and confidently knowing the way forward.