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What is a Solution to a System (Where Two Lines Cross)?
Grade Level:
Class 5
Maths, Computing, AI, Physics, Economics
Definition
What is it?
When we have two lines, a 'solution to the system' is the special point where these two lines meet or cross each other. This point has an 'x' value and a 'y' value that works for both lines at the same time.
Simple Example
Quick Example
Imagine two auto-rickshaws starting from different places and driving on different roads. If their paths cross at a specific traffic signal, that traffic signal is the 'solution' – the one point where both auto-rickshaws were at the same time.
Worked Example
Step-by-Step
Let's say Line 1 is y = x + 1 and Line 2 is y = 2x - 1. We want to find the point where they cross.
Step 1: Since both equations are equal to 'y', we can set them equal to each other: x + 1 = 2x - 1.
---Step 2: Now, we need to get all the 'x' values on one side. Subtract 'x' from both sides: 1 = 2x - x - 1, which simplifies to 1 = x - 1.
---Step 3: To find 'x', add 1 to both sides: 1 + 1 = x, so x = 2.
---Step 4: Now that we have x = 2, we can put this 'x' value into either of the original equations to find 'y'. Let's use Line 1: y = x + 1.
---Step 5: Substitute x = 2 into y = x + 1. So, y = 2 + 1, which means y = 3.
---Step 6: The point where the lines cross is (x, y) = (2, 3).
Answer: The solution is (2, 3).
Why It Matters
Finding where things cross is super important in many fields! Engineers use this to design roads and bridges, economists use it to find the perfect price for products, and computer scientists use it for graphics and game design. It helps us solve real-world problems from planning cities to making sure your favorite app works smoothly.
Common Mistakes
MISTAKE: Finding only the 'x' value and forgetting to find the 'y' value. | CORRECTION: Remember that a 'point' where lines cross always has both an 'x' coordinate and a 'y' coordinate. Always find both!
MISTAKE: Making calculation errors when moving numbers or variables from one side of the equation to the other. | CORRECTION: Always do the same operation (add, subtract, multiply, divide) to BOTH sides of the equals sign to keep the equation balanced.
MISTAKE: Substituting the 'x' value back into only one of the original equations and not checking with the other. | CORRECTION: After finding 'x' and 'y', substitute both values into BOTH original equations. If they work for both, your solution is correct!
Practice Questions
Try It Yourself
QUESTION: Find the point where the lines y = x + 3 and y = 2x + 1 cross. | ANSWER: (2, 5)
QUESTION: Two friends, Rohan and Priya, are saving money. Rohan has 100 rupees and saves 10 rupees every day. Priya has 50 rupees and saves 20 rupees every day. After how many days will they have the same amount of money? (Hint: Let 'y' be total money and 'x' be number of days). | ANSWER: After 5 days (They will both have 150 rupees)
QUESTION: Line A is given by y = 4x - 2. Line B is given by y = x + 7. What is the sum of the x and y coordinates of the solution point? | ANSWER: 13 (The solution is (3, 10), so 3 + 10 = 13)
MCQ
Quick Quiz
What does the solution to a system of two lines represent?
The length of the lines
The only point where the two lines meet
The starting point of both lines
The slope of one of the lines
The Correct Answer Is:
B
The solution to a system of two lines is the specific point (x, y) where the lines intersect or cross. It is the only point that lies on both lines.
Real World Connection
In the Real World
Imagine you are using a navigation app like Google Maps or Ola Cabs. When two routes are shown on the map and they merge or cross at a particular junction, that junction is the 'solution' point where both routes are common. This helps the app calculate the best path for you or the driver.
Key Vocabulary
Key Terms
SYSTEM OF EQUATIONS: A set of two or more equations that we solve together to find a common solution. | INTERSECTION: The point where two lines or paths cross each other. | COORDINATES: A pair of numbers (x, y) that show the exact location of a point on a graph. | VARIABLE: A letter (like 'x' or 'y') that represents an unknown number.
What's Next
What to Learn Next
Great job understanding where lines cross! Next, you can explore 'Parallel Lines' and 'Coincident Lines' to see what happens when lines don't cross, or cross everywhere. This will deepen your understanding of different types of solutions in systems of equations.
