Problem 86 - Beginning and Intermediate Algebra

Question

Decide whether the given number is a solution of the given equation. Is \(-10\) a solution of \(-x-6=-x-1 ?\)

Step-by-Step Solution

Answer

No, -10 is not a solution because substituting it does not satisfy the equation.

Step 1: Substitute "-10" for "x"

Begin by substituting the value of \(x = -10\) into the given equation \(-x - 6 = -x - 1\). This gives us:\[ -(-10) - 6 = -(-10) - 1 \] which simplifies to \[ 10 - 6 = 10 - 1 \].

Step 2: Simplify both sides of the equation

Simplify both sides of the equation from Step 1.On the left side: \(10 - 6 = 4\).On the right side: \(10 - 1 = 9\).Now we have the equation: \(4 = 9\).

Step 3: Determine if the equation holds true

Since \(4\) is not equal to \(9\), the left side does not equal the right side. Therefore, \(x = -10\) is not a solution to the equation \(-x - 6 = -x - 1\).

Key Concepts

Solving Equations

Solving equations is a fundamental aspect of algebra. It involves finding values for variables that make the equation true. In every algebraic equation, like \(- x - 6 = - x - 1\), our goal is to reach an expression where the equation is balanced. To solve such equations, the first step is often to simplify both sides of the equation. This might involve combining like terms, adding, or subtracting terms from both sides to isolate the variable. If you can make the left-hand side and the right-hand side of the equation equal by finding a certain value of the variable, then you've found a solution. Ultimately, by carefully evaluating which substitutions yield a true equation, problem-solving becomes much clearer. It’s a systematic approach that’s about balancing and isolating terms.

Substitution Method

The substitution method is a strategy used to determine if a particular number satisfies an equation. This involves putting the number directly into the equation and simplifying to see if both sides equal. When you substitute a number for the variable, you're temporarily replacing it to test if the equation holds true. For instance, in the equation \(- x - 6 = - x - 1\), we want to test if \(x = -10\) makes the equation valid.

• Begin with substituting \(x = -10\) into the equation.
• Replace every occurrence of \(x\) with \(-10\).
• Simplify both sides of the equation as you perform the calculations.

After substitution, if both sides are equal, then the number is a solution. If not, as in this case, it means \(x = -10\) is not a solution. This method effectively shows whether a potential solution is valid or not.

Solution Verification

Solution verification is a crucial step in solving algebra equations. It involves checking your answer to ensure it satisfies the original equation. It's about confirming correctness after performing calculations. The idea is to substitute your solution back into the equation and simplify both sides to see if they equal. In our example, by substituting \(x = -10\), and simplifying, it became evident that \(4 eq 9\).
Thus, the original equation is not satisfied, indicating that \(-10\) is not the solution.

• Always substitute the value back into the original equation.
• Simplify both sides step-by-step.
• Confirm equality between both sides; if they aren’t equal, re-evaluate your initial solution.

Verification helps catch errors early and ensures that your solution is reliable. It reinforces understanding and solidifies the problem-solving process by ensuring accuracy.