Problem 15 - Introductory Algebra for College Students

Question

Solve each equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. $$6 x-(3 x+10)=14$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 8\)

Step 1: Simplify the Equation

Begin by simplifying the equation. Distribute the numbers across expressions where needed. The equation simplifies to \(6x - 3x - 10 = 14\)

Step 2: Combine Like Terms

In this step, similar terms have to be combined to simplify the equation further. Hence the equation becomes: \(3x - 10 = 14\)

Step 3: Isolate the variable

The variable 'x' should be isolated by moving '10' to the other side of the equation. By adding 10 to both sides, it transposes to the other side as a positive number. Hence, the equation becomes: \(3x = 24\)

Step 4: Solve for 'x'

Lastly, solve for 'x' by dividing both sides of the equation by 3 which gives: \(x = 8\)

Step 5: Validate the Solution

By substituting 'x = 8' into the original equation, we can check if the left-hand side equals the right-hand side. Substituting 'x' in the original equation \(6x-(3x+10)\) we get: \(6*8 - (3*8 + 10) = 14\) which simplifies to \(48 - (24 + 10) = 14\) . This further gives: \(48 - 34 = 14\). Hence, the left-hand side equals the right-hand side. The solution \(x = 8\) is correct.

Key Concepts

Algebraic Problem Solving

Algebraic problem solving is a methodical process used to find unknown values or solve for variables within mathematical equations. It requires understanding the rules of algebra and applying them to manipulate equations strategically. The basic steps always involve simplifying complex expressions, combining like terms, and modifying the equation to isolate the variable of interest.

In the context of solving for an unknown value, such as in the equation provided, the first step is simplification - where any algebraic distribution or removal of unnecessary parentheses takes place. Following this by combining like terms makes the equation neater and more manageable. With these steps, we can move on to isolating the variable and finally validating the solution to ensure accuracy.

Combining Like Terms

Combining like terms is a fundamental skill in algebra that simplifies equations and makes them easier to solve. Like terms are terms that have the exact same variable raised to the same power. For instance, in the equation \(6x - 3x - 10 = 14\), \(6x\) and \(3x\) are like terms.

Why Combine Like Terms?

The purpose of combining like terms is to consolidate the equation, reducing the number of terms we need to work with. By combining the coefficients of like terms, we get \(3x - 10 = 14\). This simplification is crucial for the next steps in the equation solving process.

Isolating the Variable

Isolating the variable, often the final leg in solving an algebraic equation, involves getting the variable on one side of the equation and all other numbers on the other. Doing so provides a clear path to finding the variable's value.

For \(3x - 10 = 14\), adding 10 to both sides allows us to remove it from the side containing the variable, resulting in \(3x = 24\). With the variable \(x\) now isolated, we can easily find its value by performing an operation that renders the coefficient of the variable to one, typically through division or multiplication.

Validating Algebraic Solutions

Validating algebraic solutions is an important, often overlooked step. It ensures that the value found indeed satisfies the original equation. The process involves substituting the solution back into the original equation and checking if both sides equate.

With our solution \(x = 8\), we revisit the equation \(6x - (3x + 10)\). Performing the substitution, we should find that both sides of the equation balance, confirming that \(x = 8\) is the valid solution. This step is crucial as it not only confirms the correctness of the answer but also helps in identifying any potential mistakes that might have occurred during problem-solving.