The equation 3x + 2y = 8 has :a)Unique solutionb)No solutionc)Infinite solutionsd)Two solutionsCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

The equation can be written as,

∴ For different values of x, different values of y will exist.
∴ The above equation has many solutions.

Answer

Understanding the Equation
The equation given is 3x + 2y = 8. This is a linear equation in two variables (x and y).
Types of Solutions for Linear Equations
Linear equations can have different types of solutions:
- Unique Solution: The equation intersects at one point on the graph.
- No Solution: The lines are parallel and never intersect.
- Infinite Solutions: The lines coincide, meaning they are the same line and intersect at infinitely many points.
Analyzing the Given Equation
To determine the solution type for the equation 3x + 2y = 8, we can rearrange it into slope-intercept form (y = mx + b):
- Move 3x to the other side: 2y = -3x + 8
- Divide by 2: y = -3/2 x + 4
This shows the slope is -3/2 and the y-intercept is 4.
Infinite Solutions Explanation
Now, if we were to consider another equation which is a scalar multiple of this equation, for example, 6x + 4y = 16, we can see that:
- If we multiply the entire equation 3x + 2y = 8 by 2, we get 6x + 4y = 16.
This means both equations represent the same line, resulting in infinite solutions since every point on the line satisfies both equations.
Conclusion
Thus, the statement that the equation 3x + 2y = 8 has infinite solutions is indeed correct, as it can coincide with other similar equations.

Answer

The correct answer is indeed option 'C' - Infinite solutions. Here's why:

The given equation is a linear equation in two variables, x and y. It can be written as:

3x + 2y = 8

To understand why this equation has infinite solutions, let's try to solve it.

We can't find a unique solution for x and y because there are infinitely many combinations of x and y that satisfy the equation.

For example:

- If x = 0, then 2y = 8, so y = 4. So, (x, y) = (0, 4) is a solution.
- If x = 2, then 3(2) + 2y = 8, so 2y = 2, and y = 1. So, (x, y) = (2, 1) is another solution.
- If x = 4, then 3(4) + 2y = 8, so 2y = -4, and y = -2. So, (x, y) = (4, -2) is yet another solution.

And so on... We can find infinitely many solutions like this.

Therefore, the given equation 3x + 2y = 8 has infinite solutions.

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