What is a System of Equations with Three Variables?

S3-SA1-0360

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

A system of equations with three variables is a set of three equations, each containing three unknown values (like x, y, and z). The goal is to find one unique set of values for x, y, and z that makes all three equations true at the same time.

Simple Example

Quick Example

Imagine you go to a local kirana store and buy three different items: biscuits, namkeen, and juice. You know the total cost of buying 2 biscuits, 1 namkeen, and 1 juice. Then you know the total cost for 1 biscuit, 2 namkeen, and 1 juice. Finally, you know the total for 1 biscuit, 1 namkeen, and 2 juices. To find the individual price of each item, you would use a system of equations with three variables.

Worked Example

Step-by-Step

Let's solve this system:
Equation 1: x + y + z = 10
Equation 2: x - y + z = 6
Equation 3: 2x + y - z = 9

Step 1: Add Equation 1 and Equation 2 to eliminate 'y'.
(x + y + z) + (x - y + z) = 10 + 6
2x + 2z = 16 (Let's call this Equation 4)
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Step 2: Add Equation 2 and Equation 3 to eliminate 'y'.
(x - y + z) + (2x + y - z) = 6 + 9
3x = 15
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Step 3: Solve for 'x' using the result from Step 2.
3x = 15
x = 15 / 3
x = 5
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Step 4: Substitute the value of 'x' (which is 5) into Equation 4.
2(5) + 2z = 16
10 + 2z = 16
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Step 5: Solve for 'z'.
2z = 16 - 10
2z = 6
z = 6 / 2
z = 3
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Step 6: Substitute the values of 'x' (5) and 'z' (3) into Equation 1.
5 + y + 3 = 10
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Step 7: Solve for 'y'.
8 + y = 10
y = 10 - 8
y = 2
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Answer: So, the solution is x = 5, y = 2, and z = 3.

Why It Matters

Understanding systems of equations with three variables is crucial for solving complex problems in many fields. Engineers use them to design structures, data scientists use them to analyze large datasets, and physicists use them to understand motion and forces. These skills are essential for careers in AI/ML, data science, and engineering.

Common Mistakes

MISTAKE: Not eliminating the same variable in different pairs of equations. | CORRECTION: When you combine two equations to eliminate a variable, make sure to eliminate the *same* variable (e.g., 'y') when combining another pair of equations. This helps reduce the system to two variables.

MISTAKE: Making calculation errors when adding or subtracting equations. | CORRECTION: Double-check your addition and subtraction, especially with negative signs. A small mistake early on can lead to a completely wrong final answer.

MISTAKE: Forgetting to substitute back to find all variables. | CORRECTION: After finding one variable (e.g., x), remember to substitute its value back into simpler equations to find the other variables (y and z). Don't stop until all three unknowns are found.

Practice Questions

Try It Yourself

QUESTION: Solve for x, y, and z:
x + y + z = 6
x - y + z = 2
x + y - z = 0 | ANSWER: x = 1, y = 2, z = 3

QUESTION: Find the values of a, b, and c:
a + 2b - c = 5
2a - b + c = 0
3a + b + 2c = 10 | ANSWER: a = 1, b = 2, c = 0

QUESTION: A farmer has three types of fruits: mangoes, apples, and bananas. The total weight of 1 mango, 2 apples, and 3 bananas is 20 kg. The total weight of 2 mangoes, 1 apple, and 1 banana is 15 kg. The total weight of 1 mango, 1 apple, and 2 bananas is 13 kg. What is the weight of each fruit? | ANSWER: Mango = 5 kg, Apple = 3 kg, Banana = 3 kg

MCQ

Quick Quiz

Which of the following is NOT a system of equations with three variables?

x + y + z = 5; 2x - y + z = 3; x + 2y - z = 1

a + b = 7; b + c = 8; a + c = 9

p - q + r = 10; 2p + q - r = 5; p + q + r = 12

2x + y = 4; x - y = 1; 3x + 2y = 7

The Correct Answer Is:

Option D only has two variables (x and y) across all its equations, making it a system of equations with two variables, not three. Options A, B, and C all involve three distinct variables.

Real World Connection

In the Real World

Imagine a logistics company like Delhivery or Ecom Express planning delivery routes. They might use systems of equations with three variables to optimize how many packages can be carried by different types of vehicles (e.g., bikes, autos, vans) given their capacity, fuel consumption, and delivery time constraints for different areas. This helps them deliver parcels efficiently across Indian cities.

Key Vocabulary

Key Terms

VARIABLE: An unknown value represented by a letter (like x, y, z) | EQUATION: A mathematical statement showing two expressions are equal | SYSTEM OF EQUATIONS: A collection of two or more equations that must be solved together | ELIMINATION METHOD: A technique to solve systems of equations by adding or subtracting equations to remove a variable | SUBSTITUTION METHOD: A technique to solve systems of equations by expressing one variable in terms of another and replacing it in other equations

What's Next

What to Learn Next

Once you're comfortable with three variables, you can explore systems with more variables, which are solved using similar logical steps. You'll also learn about matrices, a powerful tool used in computer science and engineering to solve very large systems of equations quickly.