What is a System of Three Linear Equations?

S3-SA1-0120

Grade Level:

Class 7

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

Definition

What is it?

A system of three linear equations means you have three mathematical sentences, each with three different unknown values (like x, y, and z). All these equations are 'linear' because when you graph them, they would form straight lines. The goal is to find one set of x, y, and z values that makes all three equations true at the same time.

Simple Example

Quick Example

Imagine you go to a snack shop. You buy 2 samosas, 1 juice, and 3 biscuits, and pay Rs 50. Your friend buys 1 samosa, 2 juices, and 1 biscuit for Rs 40. Another friend buys 3 samosas, 1 juice, and 2 biscuits for Rs 65. If 's' is the cost of a samosa, 'j' for juice, and 'b' for biscuit, you have three equations: 2s + j + 3b = 50, s + 2j + b = 40, and 3s + j + 2b = 65. Finding the individual prices (s, j, b) is solving a system of three linear equations.

Worked Example

Step-by-Step

Let's solve a simple system:
Equation 1: x + y + z = 6
Equation 2: x - y + z = 2
Equation 3: 2x + y - z = 3

Step 1: Add Equation 1 and Equation 2 to eliminate 'y'.
(x + y + z) + (x - y + z) = 6 + 2
2x + 2z = 8 (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) = 2 + 3
3x = 5 (Let's call this Equation 5)
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Step 3: Solve Equation 5 for 'x'.
3x = 5
x = 5/3
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Step 4: Substitute the value of 'x' (5/3) into Equation 4 to find 'z'.
2(5/3) + 2z = 8
10/3 + 2z = 8
2z = 8 - 10/3
2z = 24/3 - 10/3
2z = 14/3
z = (14/3) / 2
z = 7/3
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Step 5: Substitute the values of 'x' (5/3) and 'z' (7/3) into Equation 1 to find 'y'.
x + y + z = 6
5/3 + y + 7/3 = 6
12/3 + y = 6
4 + y = 6
y = 6 - 4
y = 2
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Answer: The solution is x = 5/3, y = 2, z = 7/3.

Why It Matters

Understanding systems of equations is crucial for many real-world problems. Engineers use them to design bridges and buildings, ensuring they are stable. In data science, these systems help analyze large datasets to make predictions, like what products you might like online.

Common Mistakes

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

MISTAKE: Making calculation errors when adding or subtracting equations, especially with negative signs. | CORRECTION: Always double-check your arithmetic, especially when dealing with positive and negative numbers. Write clearly and take one step at a time.

MISTAKE: Forgetting to substitute the values back into the original equations to verify the solution. | CORRECTION: After finding x, y, and z, plug them into ALL three original equations. If they all hold true, your solution is correct.

Practice Questions

Try It Yourself

QUESTION: Is the following a system of three linear equations? 2x + 3y = 5, x + y + z = 10, 4x - y + 2z = 7 | ANSWER: No, because the first equation only has two variables (x and y), not three.

QUESTION: If x + y + z = 10 and x - y + z = 4, what is the value of 2x + 2z? | ANSWER: 14 (Add the two equations: (x+y+z) + (x-y+z) = 10+4 => 2x+2z = 14)

QUESTION: A shop sells three types of pens: A, B, and C. 2 pens of type A, 1 of B, and 1 of C cost Rs 30. 1 pen of type A, 2 of B, and 1 of C cost Rs 35. 1 pen of type A, 1 of B, and 2 of C cost Rs 25. Which equation represents the first statement? | ANSWER: 2A + B + C = 30

MCQ

Quick Quiz

Which of these is NOT a system of three linear equations?

x + y + z = 10; 2x - y + 3z = 5; x + 2y - z = 1

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

3p - q + 2r = 1; p + 2q - r = 4; p^2 + q + r = 5

m + n + k = 0; 2m - n + k = 2; m + 3n - 2k = -1

The Correct Answer Is:

Option C includes p^2, which makes the equation non-linear. Linear equations only have variables raised to the power of 1. All other options consist of three linear equations with three variables each.

Real World Connection

In the Real World

In cricket, analysts use systems of equations to figure out individual player contributions to a team's score, considering different match situations. For example, if you know the total runs scored by a team in three different matches, and how many overs different batsmen played, you can set up equations to estimate each batsman's average run rate.

Key Vocabulary

Key Terms

VARIABLE: An unknown value represented by a letter (like x, y, z) | LINEAR EQUATION: An equation where the highest power of any variable is 1, forming a straight line if graphed | SYSTEM OF EQUATIONS: A collection of two or more equations that share the same variables | SOLUTION: The set of values for the variables that makes all equations in the system true simultaneously

What's Next

What to Learn Next

Great job understanding the basics of systems of three linear equations! Next, you can explore different methods to solve these systems, like the substitution method or the elimination method in more detail. This will give you powerful tools to tackle even more complex problems.