Problem 71 - Beginning and Intermediate Algebra

Question

Decide whether the given number is a solution of the given equation. Is 2 a solution of \(x+6=x+6 ?\)

Step-by-Step Solution

Answer

Yes, 2 is a solution of the equation.

Step 1: Substitute the number into the equation

To determine if 2 is a solution, substitute 2 for the variable x in the equation. The given equation to verify is \(x + 6 = x + 6\). By substituting, it becomes \(2 + 6 = 2 + 6\).

Step 2: Simplify both sides of the equation

Simplify both sides of the substituted equation. \(2 + 6\) simplifies to 8 on both sides, giving us \(8 = 8\).

Step 3: Evaluate the equation

Compare the simplified equation \(8 = 8\) to see if both sides are equal. Since the left side equals the right side, the equation is true.

Step 4: Conclude the solution

Since substituting 2 into the equation results in a true statement \(8 = 8\), 2 is indeed a solution of the equation \(x + 6 = x + 6\).

Key Concepts

Solution Verification

When dealing with algebraic equations, verifying a solution is a crucial step. This process is about determining whether a given number, when substituted into the equation, maintains the truth of the equation. Verification involves three main steps:

• First, substitute the given number into the equation for the variable.
• Second, simplify both sides to make them easier to compare.
• Finally, check if both sides of the equation remain equal after simplifying.

If the simplified equation holds true — meaning both sides are equal — then the number is indeed a solution. For instance, in our example with the equation \(x+6=x+6\), substituting \(2\) checks if it satisfies the equation, ensuring both sides balance each other out. This confirmation offers confidence that the solution is correct, aligning with the principles of equality in mathematics.

Equation Simplification

Simplification is an essential part of working with algebraic equations. It involves reducing expressions to their simplest form, making them easier to handle and solve. In the substitution method, after inserting the number for \(x\), simplification means performing any arithmetic operations necessary.

• This includes adding, subtracting, multiplying, or dividing numbers to condense each side of the equation.
• In the example \(x+6=x+6\), substituting \(x\) with \(2\) gives us \(2+6=2+6\). Simplifying here is straightforward as it results in \(8=8\).

This step is integral because simplification brings clarity. It makes it immediately apparent whether or not the two sides of the equation are equal. Keeping equations simplified also aids in spotting mistakes and understanding the problem's fundamentals.

Substitution Method

The substitution method is a versatile approach in solving equations, particularly useful when testing potential solutions. This method entails replacing a variable with a possible numerical value to see if it holds true.

• First, choose a number you suspect could be the solution, like \(2\) in our example with \(x+6=x+6\).
• Replace every instance of the variable in the equation with this chosen number.
• Then, simplify each side to verify if the equation balances.

The goal is to determine if after substitution and simplification, the equation stands true (as both sides equalize). When this occurs, it confirms the tested number as a solution. This method efficiently tests single solutions and is part of broader problem-solving strategies in algebra, allowing students to tackle various equation types with confidence.