What is a System of Three Variables?

S3-SA1-0799

Grade Level:

Class 6

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

Definition

What is it?

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

Simple Example

Quick Example

Imagine you go to a local chai shop. You buy 3 samosas, 2 cups of chai, and 1 biscuit, and it costs Rs. 80. Your friend buys 2 samosas, 1 chai, and 2 biscuits for Rs. 65. Another friend buys 1 samosa, 3 chai, and 1 biscuit for Rs. 75. Here, the price of a samosa (x), a chai (y), and a biscuit (z) are your three unknown variables, and each purchase is an equation.

Worked Example

Step-by-Step

Let's find three numbers (x, y, z) such that:
1) x + y + z = 10
2) x - y + z = 6
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 1 and Equation 3 to eliminate 'z'.
(x + y + z) + (2x + y - z) = 10 + 9
3x + 2y = 19 (Let's call this Equation 5)

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Step 3: This is becoming complex. Let's try eliminating 'y' again using Equation 2 and Equation 3.
(x - y + z) + (2x + y - z) = 6 + 9
3x = 15

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Step 4: Solve for 'x'.
3x = 15
x = 15 / 3
x = 5

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Step 5: Substitute x = 5 into Equation 4 (2x + 2z = 16).
2(5) + 2z = 16
10 + 2z = 16
2z = 16 - 10
2z = 6
z = 3

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Step 6: Substitute x = 5 and z = 3 into Equation 1 (x + y + z = 10).
5 + y + 3 = 10
8 + y = 10
y = 10 - 8
y = 2

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Answer: So, x = 5, y = 2, and z = 3.

Why It Matters

Understanding systems of variables is like being a detective for numbers! In AI/ML, these systems help computers learn patterns, like recommending your favourite songs. Engineers use them to design bridges and buildings, ensuring they are strong and safe. Even in economics, they help predict market trends and manage resources, impacting everything from petrol prices to food availability.

Common Mistakes

MISTAKE: Forgetting to apply operations (like multiplication) to ALL terms in an equation. For example, if you multiply an equation by 2, only multiplying one side. | CORRECTION: Always multiply or divide every single term on both sides of the equation to keep it balanced.

MISTAKE: Making calculation errors when adding or subtracting positive and negative numbers. | CORRECTION: Double-check your arithmetic, especially with signs. Using a separate piece of scratch paper for calculations can help.

MISTAKE: Not substituting the found values back into the original equations to check the answer. | CORRECTION: Always substitute your final x, y, and z values into all three original equations to ensure they are all true. This confirms your solution is correct.

Practice Questions

Try It Yourself

QUESTION: Which of these is NOT a system of three variables?
A) x + y = 5, y + z = 7, x + z = 9
B) 2a + b - c = 10, a - b + 2c = 5, 3a + 2b - c = 12
C) p + q = 10, r - s = 5
| ANSWER: C) p + q = 10, r - s = 5 (This has four variables: p, q, r, s, and only two equations)

QUESTION: If x + y + z = 15 and x + y = 10, what is the value of z? | ANSWER: z = 5

QUESTION: Solve for x, y, and z:
1) x + y = 7
2) y + z = 9
3) x + z = 8
| ANSWER: x = 3, y = 4, z = 5

MCQ

Quick Quiz

A system of three variables requires how many equations to typically find a unique solution?

One equation

Two equations

Three equations

More than three equations

The Correct Answer Is:

To solve for three unknown variables, you generally need at least three independent equations. If you have fewer equations, you might have infinitely many solutions or no unique solution.

Real World Connection

In the Real World

When you use a food delivery app like Swiggy or Zomato, behind the scenes, systems of equations might be used to optimize delivery routes. They calculate the best path considering traffic (x), distance (y), and number of orders (z) to get your biryani to you quickly. Even ISRO scientists use complex systems of equations to plot rocket trajectories and satellite orbits!

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 share the same variables | SOLUTION: The specific values for the variables that make all equations in the system true simultaneously

What's Next

What to Learn Next

Great job understanding what a system of three variables is! Next, you can explore different methods to solve these systems, like substitution or elimination, which are powerful tools. Mastering these will help you tackle even more complex problems in algebra!