Solutions In Linear Equations With Multiple Unknowns

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

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This statement is false. A single linear equation with two or more unknowns may have zero, one, or infinitely many solutions depending on the coefficients in the equation.

For instance, consider the following equation: 2x + 4y = 8. This is a linear equation with two unknowns, x and y. We can solve for y as follows:

2x + 4y = 8
4y = 8 – 2x
y = 2 – 0.5x

Now we have found an expression for y in terms of x. This means that for any given value of x, there is exactly one corresponding value of y that satisfies the equation. Therefore, this equation has a unique solution for every value of x and y. In other words, it has exactly one solution and does not have infinitely many solutions.

On the other hand, consider the following equation: 2x + 4y = 10. Again, we can solve for y as follows:

2x + 4y = 10
4y = 10 – 2x
y = 2.5 – 0.5x

In this case, for any given value of x, there is exactly one corresponding value of y that satisfies the equation. However, if we try to solve for x in terms of y, we get:

2x = 10 – 4y
x = 5 – 2y

This means that for any given value of y, there is exactly one corresponding value of x that satisfies the equation. Therefore, this equation also has a unique solution for every value of x and y. In other words, it has exactly one solution and does not have infinitely many solutions.

In general, whether a linear equation with two or more unknowns has infinitely many solutions depends on the coefficients in the equation, and how they constrain the values of the unknowns. If the coefficients lead to a situation where there are fewer equations than unknowns, or where the equations contradict each other, then the system may have no solution. If the coefficients lead to a situation where some unknowns can take any value, then the system may have infinitely many solutions.

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