Question

You are in charge of buying prizes for a school contest. A one-strand rosette ribbon costs 2.00 dollars and a three-strand rosette ribbon costs 3.00 dollars . You have 10 dollars to spend. Write an equation that represents the different numbers of one-strand and three-strand rosette ribbons that you can purchase.

Short answer

The equation that represents the number of one-strand and three-strand rosette ribbons that can be purchased with 10 dollars is \( 2x + 3y \leq 10 \)

Step-by-step solution

Define the Variables

Let's denote the number of one-strand rosette ribbons as \( x \) and the number of three-strand ribbons as \( y \).

Express the Total Cost as a Linear Equation of \( x \) and \( y \)

Each one-strand rosette ribbon costs 2.00 dollars and each three-strand rosette ribbon costs 3.00 dollars. Therefore, the total cost of purchasing \( x \) one-strand ribbons and \( y \) three-strand ribbons is \( 2x + 3y \) dollars.

Write Down the Linear Equation for the Total Spending

The total spending cannot exceed 10 dollars. Therefore, the quantities \( x \) and \( y \) must satisfy the inequality \( 2x + 3y \leq 10 \).

Key concepts

Algebraic Expressions

To fully grasp what an algebraic expression is, imagine it as a phrase in mathematics that may include numbers, operators (such as +, -, *, /), and one or more variables. Variables, represented by letters such as 'x' and 'y', stand in for unknown values and can vary, hence the name 'variable'. For instance, in our exercise with the rosettes, we encounter the algebraic expression \( 2x + 3y \) which showcases the cost of 'x' number of one-strand ribbons and 'y' number of three-strand ribbons. What's fascinating about algebraic expressions is that they can model real-world situations, like calculating costs, as we see here.

When you purchase 'x' one-strand ribbons at 2.00 dollars each, this part of the expression—\( 2x \)—calculates the total cost for these ribbons. The same goes for the three-strand ribbons; 'y' ribbons at 3.00 dollars each are represented by \( 3y \). Combining these gives a comprehensive formula for the total expenditure. It's a simple yet powerful representation of real-world quantities, and tweaking the numbers can help us understand different buying scenarios.

Inequalities

Inequalities are similar to equations, but instead of indicating that two expressions are equal, they show that there's a relationship where one value is larger or smaller than another. They are written with symbols like \( < \), \( > \), \( \leq \), and \( \geq \), which stand for 'less than', 'greater than', 'less than or equal to', and 'greater than or equal to' respectively. Our school contest exercise presents the inequality \( 2x + 3y \leq 10 \), suggesting that the cost of rosette ribbons must not exceed the budget of 10 dollars.

This means you can spend less than 10 dollars, or exactly 10 dollars, but not more. In the world of math, inequalities like this one are applied to express limits and constraints in problems. Recognizing how to write such inequalities is key to solving many real-world problems as they give clear boundaries within which we must find the solution.

Systems of Equations

Think of a system of equations as a group of equations that are all interconnected, sharing common variables. Solving such a system is like discovering where different conditions meet in harmony. In typical scenarios, each equation would represent a distinct condition or relationship, and when you solve the system, you're finding the values of the variables that satisfy all the conditions simultaneously.

In the context of the ribbon-buying scenario, if we had another condition, say a specific number of ribbons needed, it would form an additional equation and, along with our inequality, create a system of equations. Although this text covers only one condition, recognizing how equations can relate to each other forms the basis for solving more complex systems where variables must satisfy multiple constraints. For beginners, visual methods such as graphing could be used to find where two or more linear relationships intersect, which would offer a visual and intuitive solution.