Solving Linear Equations with No Solution: The Role of Coefficients

In mathematics, linear equations can sometimes lack a solution. This article explores the conditions under which a linear equation has no solution, using the given equation 12x - 4a 5a - 8. We will break down the problem step-by-step to understand how coefficients, specifically the coefficient of x, determine the solvability of the equation.

Understanding the Equation

The given equation is:

12x - 4a 5a - 8

To determine if this equation has a solution, we need to analyze the coefficient of x. Specifically, we need to check if the coefficient 12 can be made zero while ensuring the equation remains valid.

Determining When the Equation Has No Solution

To have no solution, the coefficient of x must be zero. This means we need:

12 - 4a 0

Let's solve for a by rearranging the equation:

4a 12

a 12 / 4

a 3

So, we have found that the equation has no solution when a 3. We will verify this by plugging a 3 back into the original equation:

12x - 4(3) 5(3) - 8

12x - 12 15 - 8

12x - 12 7

12x 19

Since 12x 19 does not lead to a value of x that satisfies the equation, we confirm that the equation has no solution.

Generalizing the Principle

The same principle can be applied to any linear equation of the form:

mx b 0

For this equation to have no solution, the coefficient m must be zero. If m 0, then the equation reduces to:

b 0

This can only be true if b 0, which would imply that the equation is true for all values of x, giving infinitely many solutions. Therefore, the coefficient of x must be non-zero for the equation to have a unique solution.

Conclusion

In summary, the equation 12x - 4a 5a - 8 has no solution when a 3. This is because the coefficient of x is zero in this scenario, leading to a contradiction. The principle can be extended to other linear equations, where setting the coefficient of x to zero and ensuring the equation does not simplify to true for all values of x results in a scenario with no solution.