Infinitely Many Solutions MCQs Quiz | Class 9
Class IX Mathematics (Code 041), Unit II: Algebra. This quiz focuses on linear equations in two variables, specifically reasoning why they have infinitely many solutions. Answer the questions below, submit to check your score, and download the detailed PDF answer sheet.
Understanding Why Linear Equations Have Infinitely Many Solutions
In Class 9 Algebra, we study Linear Equations in Two Variables. The general form is ax + by + c = 0, where a, b, and c are real numbers, and a and b are not both zero. A fundamental property of such an equation is that it has infinitely many solutions.
Key Reasons for Infinite Solutions
- Dependent Variables: For every value assigned to ‘x’, there is a corresponding unique value of ‘y’. Since there are infinite real numbers that ‘x’ can take, there are infinite corresponding values for ‘y’.
- Geometric Representation: Geometrically, a linear equation in two variables represents a straight line. A line is a collection of points. Since a line contains an infinite number of points, and every point on the line is a solution to the equation, there are infinitely many solutions.
Example Calculation
Consider the equation 2x + y = 6. We can express y in terms of x: y = 6 – 2x.
| Value of x | Calculation (y = 6 – 2x) | Value of y | Solution (x, y) |
|---|---|---|---|
| 0 | 6 – 2(0) | 6 | (0, 6) |
| 1 | 6 – 2(1) | 4 | (1, 4) |
| 2 | 6 – 2(2) | 2 | (2, 2) |
| 3 | 6 – 2(3) | 0 | (3, 0) |
As shown, we can continue substituting values for x indefinitely to get new values for y.
Quick Revision Notes
- An equation of the form ax + by + c = 0 is a linear equation in two variables.
- A solution is an ordered pair (x, y) that satisfies the equation.
- The graph of every linear equation in two variables is a straight line.
- Linear equations in one variable (e.g., ax + c = 0) have a unique solution, whereas those in two variables have infinitely many solutions.
Extra Practice Questions
- Find four different solutions for the equation x + 2y = 6.
- Check if (2, 1) is a solution of the equation 2x + 5y = 9.
- Find the value of k if x = 2, y = 1 is a solution of 2x + 3y = k.
- Draw the graph of x + y = 7.
- Express y in terms of x for the equation 3x – 2y = 4.
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