Question

Samantha offers two different packages of yoga classes at her yoga studio. She offers two hot yoga sessions and three zero gravity yoga sessions at a total cost of $$\$ 400$$. She also offers four hot yoga sessions and two zero gravity sessions at a price of $$\$ 440$$. Samantha wants to offer a larger package for long-time clients in which the cost must exceed \(\$ 800\). If Samantha does not wish to include more than

Short answer

The least number of zero gravity yoga sessions Samantha should add to her package is 3.

Step-by-step solution

Set Up Linear Equations

Let x represent the cost of one hot yoga session and y represent the cost of one zero gravity yoga session. We have the following information: 1. Two hot yoga sessions and three zero gravity yoga sessions cost $400. 2. Four hot yoga sessions and two zero gravity sessions cost $440. We can set up two linear equations from this information: \(2x + 3y = 400\) (1) \(4x + 2y = 440\) (2)

Solve the System of Linear Equations

First, we will multiply Equation (1) by 2 so that we can eliminate x by using the method of substitution: \(4x + 6y = 800\) (3) Now we will subtract Equation (2) from Equation (3) to eliminate x: \((4x + 6y) - (4x + 2y) = 800 - 440\) \( 4y = 360\) Next, we will divide by 4 to find the cost of one zero gravity yoga session (y): \(y = 90\) Now we will substitute the value of y in Equation (1) to find the cost of one hot yoga session (x): \(2x + 3(90) = 400\) \(2x = 400 - 270\) \(x = 65\) Thus, the cost of one hot yoga session is \(65, and the cost of one zero gravity yoga session is \)90.

Create an Inequality Representing the Larger Package

Samantha wants the cost of the larger package to exceed $800, and she doesn't want more than 10 hot yoga sessions. Let h represent the number of hot yoga sessions and z represent the number of zero gravity yoga sessions. Then, we have the following inequality: \(65h + 90z > 800\) (4) We also know that \( h <= 10 \).

Solve the Inequality to Determine the Least Number of Zero Gravity Yoga Sessions

Now we can substitute the maximum allowable number of hot yoga sessions (10) into the inequality (4) and solve for the least number of zero gravity yoga sessions (z): \(65(10) + 90z > 800\) \(650 + 90z > 800\) \(90z > 150\) \(z > \dfrac{5}{2}\) Since z must be a whole number, the least number of zero gravity yoga sessions Samantha should add to her package is 3.

Key concepts

Linear Equations

Linear equations are foundational to algebra and are a common challenge on the SAT math section. They are equations of the first order, meaning they can be written in the form of ax + by = c, where x and y are variables, and a, b, and c are constants. In real-life terms, think of it as a straight line on a graph where a and b represent the slope and y-intercept, respectively.

In the context of our example with Samantha's yoga studio, each package is represented by a linear equation. The price of each yoga session (hot yoga being x and zero gravity being y) and their respective total cost dictate the two equations we established:

1. 2x + 3y = 400
2. 4x + 2y = 440

These equations form two straight lines on a graph, and the solution to the system lies at the point where these two lines intersect, indicating the price per session that satisfies both package costs.

System of Equations

When we have two or more linear equations with different variables, we can solve them together as a system of equations. The most common methods to solve these are substitution, elimination, or graphing. For the SAT and our exercise, the algebraic methods like substitution and elimination are preferred for accuracy and speed.

For Samantha's yoga packages, we used elimination by multiplying the first equation to align the coefficients of one of the variables, followed by subtracting one equation from another to find the value of one variable. After determining one variable, we can plug it back into one of the original equations to find the other, as we did to solve for the costs of hot yoga and zero gravity yoga sessions.

Inequalities

Inequalities are like equations, but instead of an equals sign, they use symbols to show the relationship of greater than (>), less than (<), greater than or equal to (≥) or less than or equal to (≤). Inequalities describe a range of possible solutions or conditions that must be met, often depicted graphically with a number line or shading on a graph.

In our SAT problem with Samantha's yoga studio, the inequality shows a condition for a special package: the total cost must be greater than \(800. Translated to the inequality 65h + 90z > 800, it signifies the combined cost of any mix of hot yoga (h) and zero gravity yoga (z) sessions exceeding \)800. This inequality is vital for understanding the conditions that Samantha wants to meet for offering her larger packages.

SAT Problem Solving

SAT problem solving involves a variety of math topics, including linear equations, systems of equations, and inequalities. When approaching these problems, it's important to carefully analyze the information given, organize it into a mathematical form, and then apply the right tools to find the solution.

In exercises like Samantha's, seeing beyond the numbers to the real-world application—packages of yoga classes—is key to understanding and solving the problem. Such exercises on the SAT assess a student's ability to apply mathematical concepts to practical scenarios. Remember, it's not just about finding the solution; it's understanding the methods and reasoning used to arrive there that is most beneficial for students.