Solution Of A Linear Equation

To determine the condition for no solution in a pair of linear equations, the condition is: Given equations are: Here: Now apply the condition: For no solution:

We are given that the pair is a solution to both equations: 1) \\ 2) \\ Substitute into the second equation: Now substitute into the first equation: Solving for :

The correct option is (B):

Correct answer: 5x + 6y = 2x + 15y

Explanation:
Let the cost of one apple be x and the cost of one orange be y.

- Siri bought 5 apples and 6 oranges. So, total cost for Siri = 5x + 6y
- Laxmi bought 2 apples and 15 oranges. So, total cost for Laxmi = 2x + 15y

Since both spent the same total amount, we equate the two expressions:
5x + 6y = 2x + 15y

This represents the correct situation.

Correct answer: 3 - y = x² + 4

Explanation:
A linear equation is an equation in which the highest power (degree) of any variable is 1. That means all variables appear to the power of 1 only, and no variable is multiplied by another.

Let’s check each option:

Option 1: 3x - 2y = y + x
Simplifying: 3x - 2y - y - x = 0 → 2x - 3y = 0 → degree = 1 (Linear Equation) ✅

Option 2: x + y = 1 → degree = 1 (Linear Equation) ✅

Option 3: 1 + 2x = y - 5
Simplifying: 2x - y = -6 → degree = 1 (Linear Equation) ✅

Option 4: 3 - y = x² + 4
Simplifying: -y = x² + 1 → y = -x² - 1 → degree = 2 (Not a linear equation) ❌

Since this equation contains x², the highest degree is 2. Hence, it is not a linear equation.

Correct answer:

Explanation:
The given equation is a linear equation in one variable:
ax + b = 0

Our goal is to find the value of x that satisfies this equation.

Step 1: Move constant term to the right-hand side
Subtract b from both sides:
ax + b - b = 0 - b
⇒ ax = -b

Step 2: Isolate x
Divide both sides by a (assuming a ≠ 0):
ax ÷ a = -b ÷ a
⇒ x = -b⁄a

So, the solution to the equation ax + b = 0 is x = -b⁄a.

To determine the condition for the pair of linear equations to have a unique solution, let's analyze the given equations:

Key Concept: A system of linear equations has a unique solution if and only if the lines represented by the equations intersect at exactly one point. This occurs when the lines are not parallel, which mathematically translates to the ratios of their coefficients not being equal. Condition for a Unique Solution: For the system to have a unique solution, the following must hold: This ensures that the lines are not parallel and thus intersect at a single point.

Final Answer: The correct condition for a unique solution is: Thus, the correct option is (1).