Understanding Solutions to Linear Equations with Two or More Unknowns

A single linear equation with two or more unknowns must always have infinitely many solutions.

This statement is not correct

This statement is not correct. A single linear equation with two or more unknowns can have a variety of possible solutions, including a unique solution, infinitely many solutions, or no solution at all.

To explain this, let’s consider a single linear equation with two unknowns as an example:

2x + 3y = 8

To determine the solution set for this equation, we need to identify the values of x and y that satisfy this equation.

1. Unique Solution:
If there is a unique solution, it means that there is only one combination of x and y that satisfies the equation. For example, if we solve the equation and find that x = 2 and y = 1, then this is the only valid solution. In this case, the equation represents a unique solution.

2. Infinitely Many Solutions:
In some cases, a linear equation with two or more unknowns may have infinitely many solutions. To understand this, we need to express the equation in a specific way. By manipulating the equation, we can identify conditions under which there are infinite solutions.

Let’s consider the equation:

3x + 6y = 18

If we divide both sides of the equation by 3, we get:

x + 2y = 6

Now, we can select any value for y, and the corresponding value of x can be determined based on the equation. For instance, if we choose y = 2, then x would be 2 as well. Similarly, if we select y = 3, x would be 0. Hence, as long as the values of x and y satisfy the equation, we can have infinitely many solutions for this equation.

3. No Solution:
Lastly, there are cases where a linear equation with two or more unknowns has no solution. This occurs when the equation is inconsistent, meaning there are no values of x and y that can satisfy the equation simultaneously. For example:

2x + 3y = 8
4x + 6y = 9

Here, there is no solution that satisfies both equations simultaneously since they represent parallel lines that will never intersect. Thus, no solution exists for this system of equations.

In summary, a single linear equation with two or more unknowns can have a unique solution, infinitely many solutions, or no solution at all. It depends on the specific coefficients and constants in the equation.

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