3 Variable System Of Equations Word Problems

Let's delve into the fascinating world of solving word problems involving systems of three-variable equations. These problems, at first glance, may seem daunting, but with a structured approach and a clear understanding of the underlying principles, they become manageable and even enjoyable to solve. Mastering these problems not only enhances your mathematical skills but also sharpens your critical thinking and problem-solving abilities.

Deciphering the Code: 3-Variable Systems of Equations

A system of three-variable equations is a set of three equations, each containing three unknown variables, typically represented as x, y, and z. The solution to this system is a set of values for x, y, and z that satisfies all three equations simultaneously. Word problems involving these systems present real-world scenarios that need to be translated into mathematical equations before they can be solved.

Why are these systems important? They model complex relationships in various fields such as:

• Engineering: Designing structures and circuits.
• Economics: Analyzing market trends and resource allocation.
• Chemistry: Balancing chemical equations.
• Computer Graphics: Creating 3D models.

The Art of Translation: Turning Words into Equations

The most challenging part of solving these word problems is often the initial translation of the given information into a set of three equations. Here's a systematic approach:

1. Read Carefully and Understand: Thoroughly read the problem statement, identifying the key pieces of information and what you are being asked to find. Highlight or underline crucial details.

2. Assign Variables: Assign variables (x, y, z) to represent the unknown quantities you need to determine. Be specific and clear about what each variable represents. For example:

   • x = the number of apples
   • y = the price of a banana
   • z = the weight of a book

3. Formulate Equations: Translate the word problem's information into three mathematical equations using the assigned variables. Look for relationships and constraints within the problem. Key phrases to watch out for:

   • "The sum of..." implies addition.
   • "Is equal to..." implies equality (=).
   • "Is greater than..." or "is less than..." implies inequality (>, <). However, in a system of equations, we aim for equalities to find a unique solution.
   • "Twice as much as..." implies multiplication by 2.

4. Double-Check: Ensure your equations accurately reflect the relationships described in the problem. Read the problem again and verify that each piece of information is correctly represented in your equations.

Solving the System: Methods and Strategies

Once you have your system of equations, you can employ several methods to solve for the variables. The two most common methods are:

• Substitution: Solve one equation for one variable in terms of the other two. Then, substitute this expression into the other two equations, effectively reducing the system to two equations with two variables. Solve this reduced system, and then back-substitute to find the value of the third variable.

• Elimination (or Linear Combination): Multiply one or more of the equations by constants so that the coefficients of one of the variables are opposites. Add the equations to eliminate that variable. Repeat this process to eliminate another variable, leaving you with a single equation with one variable. Solve for this variable and then back-substitute to find the values of the other variables.

Let's illustrate these methods with examples.

Example 1: A Fruitful Problem

A fruit vendor sells apples, bananas, and oranges. A customer buys 3 apples, 2 bananas, and 1 orange for $11. Another customer buys 1 apple, 3 bananas, and 2 oranges for $9. A third customer buys 2 apples, 1 banana, and 3 oranges for $8. What is the price of each fruit?

• Variables:

  • x = price of an apple
  • y = price of a banana
  • z = price of an orange

• Equations:

  • 3x + 2y + z = 11
  • x + 3y + 2z = 9
  • 2x + y + 3z = 8

Let's solve this using the elimination method.

1. Eliminate z from the first two equations:

   • Multiply the first equation by -2: -6x - 4y - 2z = -22
   • Add this to the second equation: -5x - y = -13 (Equation 4)

2. Eliminate z from the second and third equations:

   • Multiply the second equation by -3: -3x - 9y - 6z = -27
   • Multiply the third equation by 2: 4x + 2y + 6z = 16
   • Add these two equations: x - 7y = -11 (Equation 5)

3. Solve the system of two equations (Equation 4 and Equation 5):

   • -5x - y = -13
   • x - 7y = -11
   • Multiply the second equation by 5: 5x - 35y = -55
   • Add this to the first equation: -36y = -68
   • y = 68/36 = 17/9

4. Substitute the value of y back into Equation 5:

   • x - 7(17/9) = -11
   • x - 119/9 = -11
   • x = -11 + 119/9 = (-99 + 119)/9 = 20/9

5. Substitute the values of x and y back into the first original equation:

   • 3(20/9) + 2(17/9) + z = 11
   • 60/9 + 34/9 + z = 11
   • 94/9 + z = 11
   • z = 11 - 94/9 = (99 - 94)/9 = 5/9

Therefore, the price of an apple is $20/9, the price of a banana is $17/9, and the price of an orange is $5/9.

Example 2: The Age-Old Question

Alice, Bob, and Carol are siblings. The sum of their ages is 74. Alice is twice as old as Bob. Carol is 6 years younger than Alice. How old is each sibling?

• Variables:

  • x = Alice's age
  • y = Bob's age
  • z = Carol's age

• Equations:

  • x + y + z = 74
  • x = 2y
  • z = x - 6

Let's solve this using the substitution method.

1. Substitute x = 2y and z = x - 6 into the first equation:

   • 2y + y + (2y - 6) = 74

2. Simplify and solve for y:

   • 5y - 6 = 74
   • 5y = 80
   • y = 16

3. Substitute the value of y back into x = 2y:

   • x = 2(16)
   • x = 32

4. Substitute the value of x back into z = x - 6:

   • z = 32 - 6
   • z = 26

Therefore, Alice is 32 years old, Bob is 16 years old, and Carol is 26 years old.

Example 3: Investing Wisely

An investor has $50,000 to invest in three different accounts: a savings account, a bond fund, and a stock fund. The savings account pays 2% interest per year, the bond fund pays 5% interest per year, and the stock fund pays 8% interest per year. The investor wants to earn a total of $3,100 in interest per year. She also wants to invest twice as much in the bond fund as in the savings account. How much should she invest in each account?

• Variables:

  • x = amount invested in the savings account
  • y = amount invested in the bond fund
  • z = amount invested in the stock fund

• Equations:

  • x + y + z = 50000 (Total investment)
  • 0.02x + 0.05y + 0.08z = 3100 (Total interest)
  • y = 2x (Bond fund investment is twice the savings account investment)

Let's solve this using the substitution method.

1. Substitute y = 2x into the first two equations:

   • x + 2x + z = 50000 => 3x + z = 50000
   • 0.02x + 0.05(2x) + 0.08z = 3100 => 0.02x + 0.1x + 0.08z = 3100 => 0.12x + 0.08z = 3100

2. Solve the resulting system of two equations:

   • 3x + z = 50000
   • 0.12x + 0.08z = 3100

3. Multiply the first equation by -0.08:

   • -0.24x - 0.08z = -4000

4. Add this to the second equation:

   • (-0.24 + 0.12)x = -4000 + 3100
   • -0.12x = -900
   • x = -900 / -0.12 = 7500

5. Substitute the value of x back into y = 2x:

   • y = 2 * 7500 = 15000

6. Substitute the value of x back into 3x + z = 50000:

   • 3 * 7500 + z = 50000
   • 22500 + z = 50000
   • z = 50000 - 22500 = 27500

Therefore, the investor should invest $7,500 in the savings account, $15,000 in the bond fund, and $27,500 in the stock fund.

Tips and Tricks for Conquering Word Problems

• Practice Makes Perfect: The more problems you solve, the more comfortable you will become with the process.
• Check Your Solution: After finding the values of the variables, substitute them back into the original equations to ensure they satisfy all the conditions.
• Units Matter: Pay attention to the units of measurement in the problem and make sure your answers are in the correct units.
• Don't Be Afraid to Guess and Check (Strategically): If you are stuck, try making an educated guess for one of the variables and see if it leads you to a solution. This can help you understand the relationships between the variables.
• Break Down Complex Problems: Divide the problem into smaller, more manageable steps. Focus on translating one sentence or phrase at a time.
• Draw Diagrams or Use Visual Aids: Visual representations can often help you understand the relationships between the variables and formulate the equations.
• Seek Help When Needed: Don't hesitate to ask your teacher, classmates, or online resources for help if you are struggling.

Common Pitfalls to Avoid

• Misinterpreting the Problem: Carefully read the problem statement and make sure you understand what is being asked.
• Incorrectly Assigning Variables: Define your variables clearly and consistently.
• Making Arithmetic Errors: Double-check your calculations, especially when dealing with fractions or decimals.
• Forgetting to Check Your Solution: Always substitute your solution back into the original equations to verify that it is correct.
• Giving Up Too Easily: Solving word problems requires patience and persistence. Don't get discouraged if you don't find the solution right away.

Real-World Applications: Beyond the Textbook

The skills you develop in solving systems of three-variable equations are applicable to a wide range of real-world situations. Here are a few examples:

• Diet Planning: Determining the optimal amounts of different foods to meet specific nutritional requirements (calories, protein, vitamins).
• Mixture Problems: Calculating the amounts of different ingredients needed to create a mixture with a desired concentration.
• Network Analysis: Analyzing the flow of traffic or data in a network.
• Resource Allocation: Optimizing the allocation of resources to different projects or activities.

FAQs: Addressing Common Concerns

• Q: Can a system of three-variable equations have no solution or infinitely many solutions?

  • A: Yes, just like systems of two-variable equations, systems of three-variable equations can have no solution (inconsistent system) or infinitely many solutions (dependent system). These cases arise when the equations are contradictory or when one equation is a linear combination of the other two.

• Q: Is there always a unique solution to a system of three-variable equations?

  • A: No, there is only a unique solution if the equations are independent and consistent.

• Q: Which method (substitution or elimination) is better?

  • A: The best method depends on the specific problem. If one of the equations is easily solved for one of the variables, substitution may be easier. If the coefficients of one of the variables are already opposites or can be easily made opposites, elimination may be more efficient.

• Q: Can I use a calculator to solve these problems?

  • A: Yes, calculators with matrix capabilities can be used to solve systems of equations. However, it is important to understand the underlying concepts and be able to set up the equations correctly.

Conclusion: Mastering the Art of Problem Solving

Solving word problems involving systems of three-variable equations is a valuable skill that enhances your mathematical abilities and critical thinking. By following a structured approach, practicing regularly, and understanding the underlying concepts, you can master these problems and apply them to real-world situations. Remember to translate the words into equations carefully, choose the appropriate solution method, and always check your answers. With dedication and perseverance, you can conquer even the most challenging word problems.