Problem 18 - Elementary and Intermediate Algebra

Question

Blending Teas. Verity has recently created three custom tea blends. A 5 -oz package of Southern Sandalwood sells for \(\$ 13.15\) and contains 2 oz of Keemun tea, 2 oz of Assam tea, and 1 oz of a berry blend. A 4 -oz package of Golden Sunshine sells for \(\$ 12.50\) and contains 3 oz of Assam tea and 1 oz of the berry blend. A 6-oz package of Mountain Morning sells for \(\$ 12.50\) and contains 2 oz of the berry blend, 3 oz of Keemun tea, and 1 oz of Assam tea. What is the price per ounce of Keemun tea, Assam tea, and the berry blend?

Step-by-Step Solution

Answer

Keemun: \(2.25\), Assam: \(3.85\), Berry blend: \(0.95\)

Step 1 - Define Variables

Let the price per ounce of Keemun tea be denoted as K, the price per ounce of Assam tea as A, and the price per ounce of the berry blend as B.

Step 2 - Set Up Equations

Write equations based on the price and content of each tea blend:For Southern Sandalwood:2K + 2A + B = 13.15 For Golden Sunshine:3A + B = 12.50 For Mountain Morning:3K + A + 2B = 12.50

Step 3 - Solve the System of Equations

Firstly, solve for B from the second equation: B = 12.50 - 3AThen substitute B in the first and third equations:For Southern Sandalwood: 2K + 2A + (12.50 - 3A) = 13.15 => 2K - A = 0.65For Mountain Morning: 3K + A + 2(12.50 - 3A) = 12.50 => 3K - 5A = -12.50

Step 4 - Solve the Simultaneous Equations

By solving 2K - A = 0.65 and 3K - 5A = -12.50, use substitution or elimination methods:Multiply the first equation by 5:10K - 5A = 3.25Subtract the second equation from this new equation:7K = 15.75=> K = 2.25Substitute K back into the first equation to find A: 2(2.25) - A = 0.65=> 4.50 - A = 0.65=> A = 3.85

Step 5 - Find the Berry Blend Price

Substitute A into the equation for B:B = 12.50 - 3(3.85)=> B = 12.50 - 11.55=> B = 0.95

Step 6 - Verify the Solution

Double-check the values by substituting K, A, and B back into the original equations to ensure they hold true:For Southern Sandalwood: 2(2.25) + 2(3.85) + 0.95 = 13.15=> 13.15 = 13.15For Golden Sunshine: 3(3.85) + 0.95 = 12.50=> 12.50 = 12.50For Mountain Morning: 3(2.25) + 3.85 + 2(0.95) = 12.50=> 12.50 = 12.50

Key Concepts

Understanding Linear Equations

Linear equations are mathematical expressions of equality that involve constants and variables. They typically look like ax + by + c = 0, where x and y are variables and a, b, and c are constants.
In a system of linear equations, we deal with more than one linear equation simultaneously. These are often represented as:

• Equation 1: 2K + 2A + B = 13.15
• Equation 2: 3A + B = 12.50
• Equation 3: 3K + A + 2B = 12.50

Solving these equations helps us find out the exact values of the variables (K, A, and B).
Understanding how to set up and solve these equations is crucial for dealing with real-world problems such as the price analysis of tea blends.

Mastering the Substitution Method

The substitution method is a technique for solving systems of equations where one equation is solved for one variable, and then that expression is substituted into the other equations. Here's how it works:
First, solve for one variable in one of the equations. From our example, you can solve the second equation for B:
\[ B = 12.50 - 3A \]
Next, substitute this expression into the other equations. For the Southern Sandalwood blend, you get:
\[ 2K + 2A + (12.50 - 3A) = 13.15 \]
Simplify to get a new equation in K and A:
\[ 2K - A = 0.65 \]
Repeat this for the other equations. Solving the new simplified equations will help you find the values of all variables.
This step-by-step substitution simplifies complex problems and breaks them into manageable smaller problems.

The Elimination Method Explained

Another effective method for solving systems of equations is the elimination method. This involves adding or subtracting the equations to eliminate one of the variables.
Here's a step-by-step example:
From our simplified equations:
\[ 2K - A = 0.65 \] \[ 3K - 5A = -12.50 \]
Multiply the first equation by a suitable number so the coefficients of one of the variables are the same in both equations. Here, multiply by 5:
\[ 10K - 5A = 3.25 \]
Subtract the second equation from this new equation to eliminate A:
\[ 7K = 15.75 \]
Solve for K:
\[ K = 2.25 \]
Substituting K back into the first equation gives you A:
\[ 2(2.25) - A = 0.65 \]
\[ 4.50 - A = 0.65 \]
\[ A = 3.85 \]
Use the elimination method to simplify equations, making it easier to solve for all variables involved.

Understanding Variables in Equations

Variables are symbols used in equations to represent unknown quantities. In our example, K, A, and B are variables representing the prices per ounce of Keemun tea, Assam tea, and the berry blend, respectively.
Identifying variables correctly is crucial in setting up equations. Here's the step-by-step example from our problem:
Let K = Price per ounce of Keemun tea
A = Price per ounce of Assam tea
B = Price per ounce of the berry blend
Using these defined variables, create equations based on given data:

• Southern Sandalwood: 2K + 2A + B = 13.15
• Golden Sunshine: 3A + B = 12.50
• Mountain Morning: 3K + A + 2B = 12.50

Working with variables allows you to formulate equations and solve real-world problems involving multiple unknowns. This process provides clear answers and deepens your understanding of mathematical relationships.