Problem 87 - Beginning and Intermediate Algebra

Question

Decide whether the given number is a solution of the given equation. Is 2 a solution of \(-x-13=-15 ?\)

Step-by-Step Solution

Answer

Yes, 2 is a solution.

Step 1: Substitute the Number

Start by substituting the given number, 2, into the equation \(-x - 13 = -15\). Replace \(x\) with 2, which turns the equation into:\(-2 - 13 = -15\).

Step 2: Simplify the Equation

Simplify the left side of the equation. Calculate \(-2 - 13\):\[-2 - 13 = -15\].

Step 3: Analyze the Equation

Compare the simplified expression on the left, which is \(-15\), with the right side of the equation, which is also \(-15\).

Step 4: Conclusion

Since both sides of the equation are equal, \(2\) is indeed a solution to the equation \(-x - 13 = -15\).

Key Concepts

Substitution Method

Substitution method is a common approach in algebra for determining if a specific number satisfies an equation. It involves substituting the given number into the equation in place of the unknown variable, then simplifying the result. If the left-hand side of the equation equals the right-hand side after substitution, the number is a valid solution. This method is crucial because:

• It allows for a clear verification of potential solutions.
• It simplifies complex problems into more manageable ones by focusing on one variable at a time.

Consider the equation \(-x - 13 = -15\). To see if 2 is a solution, replace \(x\) with 2. This will lead to the equation \(-2 - 13 = -15\). If calculated correctly and both sides match, the substitution confirms the number as a solution.

Negative Numbers

Handling negative numbers might be tricky but understanding them is vital in algebra. They represent values less than zero and are symbolized by a minus sign. When performing arithmetic operations involving negative numbers:

• Subtracting a positive number is the same as adding a negative number.
• Subtracting a negative number is like adding a positive number.
• Adding two negative numbers results in a larger negative value.

In our example, the left side of the equation, after substitution, becomes \(-2 - 13\), which simplifies to \(-15\). Understanding how negative numbers behave ensures that you calculate correctly, especially when substituting in equations.

Solution Verification

Solution verification is the final step to confirm that your substitution was executed correctly. This process involves comparing both sides of your equation after simplification. Let's break down its importance:

• Verification ensures the correctness of your work.
• It helps catch errors in calculation or substitution early on.
• Develops confidence in solving algebraic equations.

After substituting 2 into the equation \(-x - 13 = -15\) and simplifying, you find that both sides equal \(-15\). This equality indicates that 2 is the correct solution. Always remember, a solution satisfies the equation if the left-hand and the right-hand results match after substitution.