Problem 72 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

Yes, 10 is a solution.

Step 1: Identify the Equation

The given equation is \(x + 6 = x + 6\).

Step 2: Substitute the Value

Plug the number 10 into the variable \(x\) in the equation, making it \(10 + 6 = 10 + 6\).

Step 3: Simplify Both Sides

Calculate each side: \(10 + 6 = 16\) and \(10 + 6 = 16\). So both sides simplify to 16.

Step 4: Check Equality

Compare the simplified expressions from both sides. Since \(16 = 16\), both sides are equal.

Step 5: Conclusion

Since both sides of the equation are equal after substitution, 10 is indeed a solution to the equation \(x + 6 = x + 6\).

Key Concepts

Substitution Method

The substitution method is a fundamental concept in solving equations. It involves replacing the variable in the equation with a specific numerical value to check if it satisfies the equation.
For example, when we are given the equation \(x + 6 = x + 6\), and asked if a specific number, such as 10, is a solution, we use the substitution method to find out.
Here's how it works:

• Take the given equation \(x + 6 = x + 6\).
• Substitute the specific number, in this case, 10, in place of \(x\). This changes the equation to \(10 + 6 = 10 + 6\).

By replacing \(x\) with 10, we are checking if the equation holds true with this substitution. If it does, then 10 is indeed a solution. This method can be applied to any similar problem where you need to verify if a given number is a solution to an equation.

Simplifying Equations

Simplifying equations is an important step that involves performing arithmetic operations on both sides of an equation to reduce it to a simpler form.
In our given problem, after substituting 10 for \(x\), we have the equation \(10 + 6 = 10 + 6\).

• On the left side, \(10 + 6\) simplifies to 16.
• On the right side, \(10 + 6\) also simplifies to 16.

This simplification shows that both sides of the equation are transformed into the same number, 16, confirming that the equation is balanced. Simplifying both sides ensures that we have clear and straightforward values to compare in the next step. It makes the equation easier to interpret and solve.

Checking Equality

Checking equality is the final step in determining if a number is a solution to an equation.
After we simplify the equation on both sides, we need to verify that these simplified expressions are equal. In our example, we simplified both sides to 16.

• After substituting and simplifying, we found \(16 = 16\).
• This proves that the equation is true.

Thus, checking equality confirms our answer. If the simplified numbers on both sides of the equation are the same, then the substituted number is a solution. This final step assures us that 10 is truly a solution to the equation \(x + 6 = x + 6\). It's a reassuring process that helps to double-check our solution before concluding.