Problem 60 - Algebra 1

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. Graph the linear function \(8 x+5 y=600\)

Step-by-Step Solution

Answer

The linear function \(y = 120 - 1.6x\) can be graphed starting from the y-intercept (0, 120) and plotting a line with slope -1.6. The line represents possible combinations of adult and student tickets that could be sold to reach the fundraising goal.

Step 1: Rewrite the equation

The equation must be rewritten in y = mx + b form. That is, isolate y. To do so, subtract 8x from both sides of the equation \(8 x+5 y = 600\), which gives us \(5y = 600 - 8x\). Then, divide all quantities by 5, so we obtain the equation: \(y = 120 - 1.6x\). So m (slope) = -1.6 and b (y-intercept) = 120.

Step 2: Graph the linear function

Start by plotting the y-intercept (0, 120) on the graph. Since the slope is -1.6, this means that for every 1 unit increase in x, y will decrease by 1.6 units. Using this concept, plot a second point by moving 1 unit to the right and 1.6 units down from the y-intercept. Draw a line through these points to represent the linear function.

Step 3: Interpret the graph

The x-intercept of the line, the point where y=0, represents the number of adult tickets that could be sold to reach the goal if no student tickets were sold. The y-intercept, where x=0, represents the number of student tickets that could be sold to reach the goal if no adult tickets were sold. Any point on the line represents possible combinations of adult and student tickets that would reach the goal.

Key Concepts

Graphing Linear Equations

Graphing a linear equation like \(8x + 5y = 600\) involves first transforming it into a more familiar slope-intercept form, \(y = mx + b\). This allows you to easily identify key properties of the line, such as its slope and y-intercept. By isolating \(y\), we convert our equation to \(y = 120 - 1.6x\). Here, \(m\) represents the slope and is equal to \(-1.6\), while \(b\) is the y-intercept at 120.

These elements are crucial because:

• The slope, \(m = -1.6\), tells us the rate at which \(y\) decreases for every increment in \(x\).
• The y-intercept, \(b = 120\), shows the point where the line crosses the y-axis, representing all scenarios where \(x = 0\).

To graph, start at the y-intercept point \((0, 120)\). For every unit you move right horizontally, go 1.6 units down vertically. Drawing a line through these points gives a visual look at how adult and student ticket combinations affect total sales.

Systems of Equations

Systems of equations involve solving for multiple variables interacting within multiple equations. In this context, the equation \(8x + 5y = 600\) represents one constraint on our ticket sales system, describing how adult and student ticket sales generate revenue.

Consider other potential equations that might exist in a complex system:

• Budget limits or costs associated with producing the play, possibly creating another equation.
• Maximum seating capacity or restrictions, impacting the number of possible ticket sales.

However, our current context only deals with one equation, focusing on the revenue goal. Solving such systems typically requires methods like graphing, substitution, or elimination to find where multiple equations intersect, revealing potential solutions.

Intercepts

Intercepts are critical in understanding linear equations graphically and numerically. They help us see where our line crosses the axes, indicating scenarios where one variable reaches zero while the other is not.

• The x-intercept occurs when \(y=0\). Substitute \(y=0\) in the equation \(8x + 5y = 600\) to find \(x\). This shows the maximum number of adult tickets sold if no student tickets are sold.
• The y-intercept is the point \((0, 120)\), explained as the total scenario of student ticket sales when no adult tickets are sold.

By analyzing intercepts, we can decipher specific outcomes achievable under different conditions and graphically represent this understanding on a coordinate plane. This approach simplifies comprehending the balance necessary to achieve the desired total sales, illustrating all possible ticket combinations that meet the financial goal.