Problem 126 - Prealgebra and Introductory Algebra

Question

Is \(-\frac{1}{6}\) a solution of the equation \(6 x=1 ?\)

Step-by-Step Solution

Answer

No, \(-\frac{1}{6}\) is not a solution of the equation \(6 x = 1\).

Step 1: Substitution

Replace variable \(x\) with \(-\frac{1}{6}\) in the equation to obtain \(6 \cdot -\frac{1}{6} = 1\)

Step 2: Simplification

Simplify the left side of the equation to get \(-1 = 1\)

Step 3: Comparison

Check if the left side of the equation equals the right side. -1 does not equal 1

Key Concepts

Substitution Method

The substitution method is a fundamental technique used in algebra to find the value of variables in equations. It involves replacing the variable with a given number or expression. This method can be particularly useful when you want to verify if a certain number is a solution to an equation.

Consider the original problem where you have the equation \(6x = 1\) and you want to know if \(-\frac{1}{6}\) is a solution. To apply the substitution method, you replace every instance of the variable \(x\) in the equation with \(-\frac{1}{6}\). Essentially, you're substituting the variable with a potential solution to see if it balances the equation.

Substitution can also be used to solve systems of equations by expressing one variable in terms of the others and substituting it into another equation, thereby reducing the number of variables and equations you're working with. Remember, when using the substitution method, always perform the substitution with care to avoid any mistakes that could lead to incorrect conclusions about whether a value is truly a solution to the given equation.

Equation Simplification

Equation simplification is the process of reducing an equation to its simplest form, making it easier to solve or understand. It involves combining like terms, distributing multipliers, and following the order of operations: parentheses, exponents, multiplication and division, and addition and subtraction (PEMDAS).

For the problem at hand, after substituting \(-\frac{1}{6}\) for \(x\), you would simplify the equation. In this case, simplification means multiplying \(6\) by \(-\frac{1}{6}\). Paying attention to the negative sign is crucial since it affects the multiplication result. When simplified, the product of \(6\) and \(-\frac{1}{6}\) is \(-1\), which demonstrates the importance of precise and careful simplification. Simplification should always lead to an equation or an expression that's easier to evaluate or compare.

Numerical Solution

A numerical solution of an equation represents a specific number that satisfies the equation, meaning that when the variable of an equation is replaced with this number, the equation holds true with both sides equal. Finding numerical solutions is the ultimate goal in solving equations.

In our example, after the substitution and simplification steps, we reach the stage of determining the numerical solution. By assessing whether the left side of the simplified equation \(-1\) is equal to the right side \(1\), we determine the numerical validity of the potential solution. Since \(-1\) does not equal \(1\), we conclude that \(-\frac{1}{6}\) is not a numerical solution to the equation \(6x = 1\). The pursuit of a numerical solution often involves trial and error, especially when dealing with more complex equations, and is not limited to finding a single solution, as some equations have multiple valid numerical solutions.