91Ó°ÊÓ

When solving linear equations, understanding the distributive property is an essential first step. This mathematical principle is applied when a number outside the parentheses needs to be multiplied by each term inside the parentheses. It is usually in the form of \(a(b + c)\), which results in \(ab + ac\).

For example, in the exercise \(x - 3(2x + 3) = 8 - 5x\), the distributive property is used to expand \(-3(2x + 3)\). This multiplication yields \(-6x - 9\), simplifying the equation to \(x - 6x - 9 = 8 - 5x\). This property ensures that multiplication is correctly distributed across the terms, preventing errors in further steps of solving the equation.

Once the distributive property is applied, the next step in solving linear equations involves combining like terms. This process helps in simplifying the equation by adding or subtracting the coefficients of variables that are the same, and combining constants.

In the given problem, after distribution, we combine the terms with \(x\) on the left side yielding \(-5x\), and we also combine the constants. This simplification leads to the equation \(-5x - 9 = 8 - 5x\). Combining like terms is crucial for reducing the complexity of an equation, making it easier to understand and solve.

In certain cases, after following the steps of solving an equation, we arrive at a statement that is untrue and impossible, indicating that the equation has no solution. This scenario occurs when the variables cancel out, and we are left with a false numerical statement.

For example, when we add \(5x\) to both sides of the simplified equation \(-5x - 9 = 8 - 5x\), the variables eliminate each other, leaving us with \(-9 = 8\). Since this statement is obviously false, it indicates there is no possible value of \(x\) that could satisfy the original equation, hence the equation has 'no solution'. Understanding this concept is important as it signifies the limits of certain equations and the need to recognize situations where no solution exists.