Problem 71 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

No, \(-2\) is not a solution because both sides aren't equal.

Step 1: Substitute the Number into the Equation

First, substitute \( x = -2 \) into the equation \( -x + 6 = -x - 1 \). This gives us: \(-(-2) + 6 = -(-2) - 1\).

Step 2: Simplify Both Sides of the Equation

Now simplify both sides of the equation. The left side: \(-(-2) + 6 = 2 + 6 = 8\). The right side: \(-(-2) - 1 = 2 - 1 = 1\).

Step 3: Compare Both Sides of the Equation

Compare the simplified form of both sides. The left side equals \(8\) and the right side equals \(1\). Since \(8 eq 1\), the number \(-2\) is not a solution.

Key Concepts

Substitution Method

The substitution method is a handy tool for solving equations, especially when it comes to determining if a specific number is a solution to a given equation. Let's walk through how this method works.

The idea is simple: "substitute" simply means to replace. You take the number that is being tested and substitute it, or plug it, into the variable of the equation. In our example, we have the equation \(-x + 6 = -x - 1\) and are trying to see if \(x = -2\) is a solution.

Here's what you do:

• Identify the variable in the equation. Here, it's \(x\).
• Take the given number, \(-2\) in this case, and replace \(x\) with it.
• Rewrite the equation using \(-2\) wherever \(x\) appears. You'll get \(-(-2) + 6 = -(-2) - 1\).

This new equation will help us see if the original statement holds true once \(x\) is replaced with the given number.

Simplifying Expressions

Once you've substituted the number into the equation, the next step is to simplify both sides. Simplifying expressions means reducing them to their simplest form. In our example, we start with the equation \(-(-2) + 6 = -(-2) - 1\).

Breaking it down:

• The left side: Calculate \(-(-2)\). The double negative turns into a positive, so \(-(-2)\) becomes \(2\). Add \(6\) to \(2\), resulting in \(8\).
• The right side: Again, calculate \(-(-2)\) which becomes \(2\). Then, subtract \(1\) to get \(1\).

After simplifying, we have \(8\) on one side and \(1\) on the other. Note that simplifying is crucial as it provides a clearer picture of whether both sides are equal or not.

Checking Solutions

The final step is to determine if the number you substituted is indeed a solution to the equation. This involves comparing the results you obtained after simplifying. For a number to be a solution, both sides of the equation should match after you substitute and simplify.

In the example we've worked through, after substituting and simplifying, we found:

• The left side equals \(8\).
• The right side equals \(1\).

Since \(8 eq 1\), the equation does not hold true. Therefore, \(-2\) is not a solution.

While checking solutions, it's always about ensuring both sides of the equation are equal after you've worked through substitution and simplification. If they aren't, the number is not a solution. This method is reliable and should always be part of your process when solving equations.