Problem 98 - Algebra 1: Concepts and Skills

Question

You are buying a combination of irises and lilies for a flower arrangement. The irises are \(\$4\) each and the lilies are \(\$3\) each. You spend \(\$50\) for an arrangement of 15 flowers. How many of each type of flower did you buy?

Step-by-Step Solution

Answer

You bought 5 irises and 10 lilies.

Step 1: Establish The Variables

First, establish the variables that are needed. Let \(x\) represent the number of irises and \(y\) the number of lilies.

Step 2: Formulate The Equations

Next, formulate the equations based on the given information. From the total number of flowers, we get the equation \(x + y = 15\). From the total cost of the flowers, we get the equation \(4x + 3y = 50\).

Step 3: Solve The System Of Equations

The system of equations can be solved using several methods, such as substitution, elimination or graphical method. In this case, we will use the substitution method. Subtract \(y\) from both sides of the first equation to get \(x = 15 - y\). Substitute \(x\) in the second equation with \(15 - y\), so it becomes \(4(15-y)+ 3y = 50\). Simplify this to get \(y = 10\). Substitute \(y =10\) into the first equation to get \(x = 15 - 10\), which simplifies to \(x = 5\). So, \(y = 10\) and \(x = 5\).

Step 4: Interpret The Solution

The values obtained for \(x\) and \(y\) represent the number of irises and lilies bought respectively. So, 5 irises and 10 lilies were bought.

Key Concepts

Linear Equations

Linear equations are fundamental in solving everyday problems. They involve equations where variables are raised only to the first power and appear in a straight line when graphed. In our flower purchasing scenario, we use the concept of linear equations to model the relationship between two different types of flowers purchased.
By letting \(x\) represent the number of irises and \(y\) the number of lilies, we constructed two linear equations:

• From the total number of flowers: \(x + y = 15\)
• From the total cost of the flowers: \(4x + 3y = 50\)

These equations collectively form a system of linear equations that can be solved to find out precisely how many irises and lilies were bought. Linear equations simplify complex word problems by breaking down relationships into manageable parts.

Substitution Method

The substitution method is a powerful algebraic tool used for solving systems of equations. Here, we choose one equation and solve for one variable in terms of the others, then substitute this into another equation. This method helps to simplify the problem effectively, step-by-step.
In our example, we first solved the equation \(x + y = 15\) for \(x\), yielding \(x = 15 - y\). This expression for \(x\) was then substituted into the second equation \(4x + 3y = 50\).

• Replaced \(x\) with \(15 - y\) to get: \(4(15 - y) + 3y = 50\)
• Simplifying gives \(60 - 4y + 3y = 50\)
• This solves to \(y = 10\)

Once \(y\) was determined, we substituted it back into the original \(x = 15 - y\) to find \(x = 5\). The substitution method clearly demonstrated how to find a solution to the system systematically.

Word Problems

Word problems involve real-world scenarios where mathematics helps us find solutions. Initially, these problems might seem challenging as they combine comprehension skills with mathematical skills. However, they become quite manageable once broken down into steps.
In our flower arrangement example, word problems required transforming a practical scenario into mathematical equations. Key steps in tackling such problems include:

• Reading the problem carefully to understand what is being asked
• Identifying and defining variables for the unknowns in the problem
• Writing equations that model the relationships described in the problem

These steps help translate narrative descriptions into actionable math problems. Breaking it down like this makes solving the overall problem less daunting and more logical. In applications like buying a combination of items (flowers in our example), word problems assist in determining optimal solutions efficiently.