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What is the Solution of a Linear Equation in Two Variables?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
A solution of a linear equation in two variables is a pair of values (one for each variable) that makes the equation true. When you substitute these values into the equation, the left side becomes equal to the right side.
Simple Example
Quick Example
Imagine you buy some pens and pencils. If 2 pens and 3 pencils cost Rs 20, we can write this as 2x + 3y = 20 (where x is pen price and y is pencil price). A solution could be (x=4, y=4), because 2(4) + 3(4) = 8 + 12 = 20. So, a pen costing Rs 4 and a pencil costing Rs 4 is one possible solution.
Worked Example
Step-by-Step
Let's find if (x=5, y=2) is a solution for the equation 3x + 2y = 19.
1. Write down the equation: 3x + 2y = 19.
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2. Substitute the value of x (which is 5) into the equation: 3(5) + 2y = 19.
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3. Substitute the value of y (which is 2) into the equation: 3(5) + 2(2) = 19.
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4. Multiply the numbers: 15 + 4 = 19.
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5. Add the numbers on the left side: 19 = 19.
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6. Since the left side (19) is equal to the right side (19), the values (x=5, y=2) are indeed a solution to the equation.
Answer: Yes, (x=5, y=2) is a solution.
Why It Matters
Understanding solutions helps engineers design circuits and scientists model how things change. Data scientists use these concepts to find patterns, and economists predict market trends. It's a foundational skill for future careers in technology and research!
Common Mistakes
MISTAKE: Swapping x and y values when substituting. For example, using x=2 and y=5 for (5,2). | CORRECTION: Always remember the order: (x-value, y-value). The first number is for 'x', the second for 'y'.
MISTAKE: Making calculation errors after substituting. Forgetting to multiply or adding incorrectly. | CORRECTION: Double-check your arithmetic carefully after substituting the values. Take your time with calculations.
MISTAKE: Assuming there is only one solution. | CORRECTION: A single linear equation in two variables usually has infinitely many solutions, not just one. (5,2) is just ONE possible solution, not the only one.
Practice Questions
Try It Yourself
QUESTION: Is (x=3, y=1) a solution for the equation 4x + y = 13? | ANSWER: Yes
QUESTION: Is (x=2, y=5) a solution for the equation 5x - 2y = 0? | ANSWER: No
QUESTION: If (x=1, y=k) is a solution for the equation 7x + 3y = 13, what is the value of k? | ANSWER: k = 2
MCQ
Quick Quiz
Which of the following pairs is a solution for the equation 2x + y = 10?
(3, 2)
(4, 2)
(2, 4)
(5, 1)
The Correct Answer Is:
B
For option B, if x=4 and y=2, then 2(4) + 2 = 8 + 2 = 10. This makes the equation true. The other options do not satisfy the equation.
Real World Connection
In the Real World
Imagine you're a shopkeeper in India trying to manage your inventory. You sell 'atta' (flour) and 'dal' (lentils). If you know the total weight of atta and dal you sold today, and the total money you earned, you can use linear equations to find out how much of each item you sold, helping you restock efficiently.
Key Vocabulary
Key Terms
VARIABLE: A letter (like x or y) representing an unknown number. | EQUATION: A statement that two mathematical expressions are equal. | SUBSTITUTE: To replace a variable with a specific number. | LINEAR EQUATION: An equation where the highest power of the variable is 1.
What's Next
What to Learn Next
Great job understanding solutions! Next, you'll learn about 'Graphing Linear Equations in Two Variables'. This will show you how all these solutions can be seen as points forming a straight line on a graph, which is super cool!
