Problem 85 - Beginning and Intermediate Algebra

Question

Decide whether the given number is a solution of the given equation. Is \(-2\) a solution of \(-x+6=-x-1 ?\)

Step-by-Step Solution

Answer

No, \(x = -2\) is not a solution.

Step 1: Understand the equation

The given equation is \(-x + 6 = -x - 1\). We are tasked with determining if \(x = -2\) is a solution to this equation.

Step 2: Substitute the number into the equation

Substitute \(x = -2\) into the equation: \[-(-2) + 6 = -(-2) - 1\].

Step 3: Simplify both sides of the equation

Calculate the expression on both sides. The left side becomes \[2 + 6 = 8\], and the right side becomes \[2 - 1 = 1\].

Step 4: Compare both sides of the equation

Compare the results from step 3. The left side of the equation is 8, and the right side is 1. Since they are not equal, \(x = -2\) is not a solution of the equation.

Key Concepts

Solution of Equation

A solution of an equation is a value that makes the equation true when substituted in place of the variable. In our exercise, the equation given is \(-x + 6 = -x - 1\).
To determine if a particular number, like \(x = -2\), is a solution, we substitute it into the equation and simplify both sides.
If both sides are equal after substitution, then that number is a solution. If not, it is not.
In this case, our task is to check the validity of \(x = -2\) as a solution.

Substitution Method

The substitution method involves replacing the variable in the equation with a given number to test if it results in a true statement. In our example, we substitute \(x = -2\) into the equation \\(-x + 6 = -x - 1\) to see if it balances.
Here's how it works: you take every occurrence of \(x\) in the equation and replace it with \(-2\).

• For the left side: \[-(-2) + 6\]
• For the right side: \[-(-2) - 1\]

By performing these substitutions, we prepare the equation for the next step: simplification.

Simplifying Equations

Simplifying an equation means performing mathematical operations to make it easier to compare both sides.
Once \(x = -2\) is substituted, simplify by carrying out the arithmetic operations. This involves handling negatives, addition, or subtraction.
For the left side, \[-(-2) + 6\], calculate as follows: 1. The negation of \(-2\) is \(2\).2. Then add \(6\) giving \(8\).

• Thus, the left simplifies to \(8\).

For the right side, \[-(-2) - 1\], simplify similarly: 1. Negation of \(-2\) is \(2\).2. Subtract \(1\) resulting in \(1\).

• So the right simplifies to \(1\).

Now both sides are simplified and ready for comparison.

Comparing Expressions

After substituting and simplifying the expressions on both sides of the equation, the final step is to compare these results. Successful comparison determines if \(x = -2\) is a solution.
In our equation, after substitution and simplification, we have:

• Left side = \(8\)
• Right side = \(1\)

Since \(8 eq 1\), these are not equal, which means \(x = -2\) is not a solution.
The equality check is crucial: if both sides had matched, \(-2\) would have been a solution, confirming it satisfies the original equation.