What is a Solution to a Linear Equation?

S6-SA1-0004

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

A solution to a linear equation is a value (or set of values) for the variable(s) that makes the equation true. When you substitute these values into the equation, both sides of the equation become equal.

Simple Example

Quick Example

Imagine you bought 2 samosas and paid 20 rupees. If 'x' is the cost of one samosa, the equation is 2x = 20. The solution is x = 10, because if one samosa costs 10 rupees, then 2 * 10 = 20, which is true.

Worked Example

Step-by-Step

Let's find the solution for the linear equation: 3x + 5 = 17.
---Step 1: Our goal is to get 'x' by itself on one side of the equation. First, subtract 5 from both sides to isolate the term with 'x'.
3x + 5 - 5 = 17 - 5
3x = 12
---Step 2: Now, 'x' is being multiplied by 3. To get 'x' alone, divide both sides of the equation by 3.
3x / 3 = 12 / 3
x = 4
---Step 3: To check our answer, substitute x = 4 back into the original equation.
3(4) + 5 = 17
12 + 5 = 17
17 = 17
---Since both sides are equal, our solution is correct. The solution to the equation 3x + 5 = 17 is x = 4.

Why It Matters

Understanding solutions to linear equations is super important! Engineers use them to design bridges and circuits, while data scientists use them in AI to make predictions. Even doctors use them to calculate medicine dosages, helping improve lives every day.

Common Mistakes

MISTAKE: Forgetting to perform the same operation on both sides of the equation. For example, subtracting 5 from the left side but not the right. | CORRECTION: Whatever you do to one side of the equation (add, subtract, multiply, divide), you MUST do to the other side to keep the equation balanced.

MISTAKE: Confusing addition/subtraction with multiplication/division. For example, in 3x = 12, students might subtract 3 instead of dividing by 3. | CORRECTION: Always use the inverse operation to isolate the variable. If a number is added, subtract it; if multiplied, divide it.

MISTAKE: Incorrectly combining terms, especially with negative numbers. For example, 5 - 7 = 12. | CORRECTION: Pay close attention to signs when adding or subtracting numbers. Remember rules for operations with positive and negative integers.

Practice Questions

Try It Yourself

QUESTION: Find the solution for the equation: x + 7 = 15. | ANSWER: x = 8

QUESTION: Solve for y: 4y - 6 = 18. | ANSWER: y = 6

QUESTION: The cost of 3 pens and 1 notebook is Rs 70. If the notebook costs Rs 25, what is the cost of one pen? (Let 'p' be the cost of one pen). | ANSWER: p = 15

MCQ

Quick Quiz

Which of the following values is a solution to the equation 2x - 3 = 7?

x = 2

x = 3

x = 5

x = 7

The Correct Answer Is:

If x = 5, then 2(5) - 3 = 10 - 3 = 7. This makes the equation true. Options A, B, and D do not make the equation true when substituted.

Real World Connection

In the Real World

When you use a ride-sharing app like Ola or Uber, the fare calculation often involves linear equations based on distance and time. For example, a base fare plus a certain amount per kilometer. Finding the 'solution' helps determine how much you pay for a specific distance.

Key Vocabulary

Key Terms

VARIABLE: A symbol (usually a letter like x, y) representing an unknown value | EQUATION: A mathematical statement showing two expressions are equal | CONSTANT: A fixed value that does not change | INVERSE OPERATION: An operation that undoes another operation (e.g., addition and subtraction)

What's Next

What to Learn Next

Great job understanding solutions! Next, you can explore 'Solving Linear Equations with Two Variables'. This will help you solve more complex real-world problems where multiple unknowns are involved, like planning a family budget!