Systems Of Equations Word Problems: Answers & Solutions

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11-16-2024

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Systems of equations play a crucial role in mathematics, particularly in solving real-world problems. In this article, we will explore various word problems involving systems of equations, showcasing how they can be solved and providing clear answers and solutions. 🌟

Understanding Systems of Equations

A system of equations is a set of two or more equations with the same variables. The goal is to find the values of the variables that satisfy all the equations in the system simultaneously. Systems of equations can be solved using several methods, including:

• Graphing 📉
• Substitution 🔄
• Elimination ❌

Types of Systems

1. Consistent System: Has at least one solution.
2. Inconsistent System: Has no solution.
3. Dependent System: Has infinitely many solutions.

Word Problems Involving Systems of Equations

Let’s delve into some examples of word problems and their solutions to better understand how systems of equations work in practice. 💼

Example 1: The Coin Problem

Problem Statement: Anna has a total of 75 coins consisting of nickels and dimes. If the total value of the coins is $5.25, how many nickels and dimes does Anna have?

Step 1: Define Variables

• Let ( n ) = number of nickels
• Let ( d ) = number of dimes

Step 2: Set Up the Equations

1. ( n + d = 75 ) (Equation 1: Total number of coins)
2. ( 0.05n + 0.10d = 5.25 ) (Equation 2: Total value of coins)

Step 3: Solve the System of Equations Using the substitution method, we can solve these equations step by step.

From Equation 1, we can express ( d ) in terms of ( n ): [ d = 75 - n ]

Substituting ( d ) in Equation 2: [ 0.05n + 0.10(75 - n) = 5.25 ]

Now simplify: [ 0.05n + 7.50 - 0.10n = 5.25 ] [ -0.05n + 7.50 = 5.25 ] [ -0.05n = 5.25 - 7.50 ] [ -0.05n = -2.25 ] [ n = \frac{-2.25}{-0.05} = 45 ]

Now substitute ( n ) back to find ( d ): [ d = 75 - 45 = 30 ]

Final Answer: Anna has 45 nickels and 30 dimes. 🪙

Example 2: The Age Problem

Problem Statement: The sum of the ages of two siblings, Tom and Jerry, is 30 years. If Tom is three years older than Jerry, how old are they?

Step 1: Define Variables

• Let ( t ) = Tom's age
• Let ( j ) = Jerry's age

Step 2: Set Up the Equations

1. ( t + j = 30 ) (Equation 1: Total age)
2. ( t = j + 3 ) (Equation 2: Tom is older)

Step 3: Solve the System of Equations Substituting Equation 2 into Equation 1: [ (j + 3) + j = 30 ] [ 2j + 3 = 30 ] [ 2j = 30 - 3 ] [ 2j = 27 ] [ j = \frac{27}{2} = 13.5 ]

Now substitute ( j ) back to find ( t ): [ t = j + 3 = 13.5 + 3 = 16.5 ]

Final Answer: Tom is 16.5 years old, and Jerry is 13.5 years old. 🎂

Example 3: The Mixture Problem

Problem Statement: A chemist has a solution that is 20% acid and another solution that is 50% acid. How many liters of each solution must be mixed to obtain 10 liters of a solution that is 30% acid?

Step 1: Define Variables

• Let ( x ) = liters of 20% solution
• Let ( y ) = liters of 50% solution

Step 2: Set Up the Equations

1. ( x + y = 10 ) (Equation 1: Total volume of the mixture)
2. ( 0.20x + 0.50y = 0.30(10) ) (Equation 2: Total acid content)

Step 3: Solve the System of Equations From Equation 1: [ y = 10 - x ]

Substituting into Equation 2: [ 0.20x + 0.50(10 - x) = 3 ] [ 0.20x + 5 - 0.50x = 3 ] [ -0.30x + 5 = 3 ] [ -0.30x = 3 - 5 ] [ -0.30x = -2 ] [ x = \frac{-2}{-0.30} \approx 6.67 ]

Now find ( y ): [ y = 10 - 6.67 \approx 3.33 ]

Final Answer: To obtain 10 liters of 30% acid solution, mix approximately 6.67 liters of the 20% solution and 3.33 liters of the 50% solution. ⚗️

Key Takeaways

• Systems of equations are essential for solving real-world problems involving relationships between quantities.
• It is crucial to correctly define variables, set up the equations, and choose the right method for solving them.
• Mastering the techniques for solving systems of equations can lead to better problem-solving skills in various disciplines, including business, science, and engineering.

Conclusion

Understanding and solving systems of equations is a valuable skill that applies to numerous real-life situations. By practicing various word problems, you can enhance your mathematical abilities and make sense of complex relationships through equations. Remember, whether you're dealing with coins, ages, or mixtures, the key lies in breaking down the problem into manageable parts and using the right methods to find solutions. Keep practicing, and soon enough, you'll become a pro at systems of equations! 💪