Problem 68 - Beginning and Intermediate Algebra

Question

Decide whether the given number is a solution of the given equation. Is 2 a solution of \(4 x+2=x+8 ?\)

Step-by-Step Solution

Answer

Yes, 2 is a solution.

Step 1: Substitute the number into the equation

Substitute \(x = 2\) into the equation \(4x + 2 = x + 8\). This results in the expression: \(4(2) + 2\).

Step 2: Simplify the left side of the equation

Simplify the expression obtained in Step 1: \(4(2) + 2\). Calculate \(4 \times 2 = 8\), then add 2 to get 10. So, the left side of the equation simplifies to 10.

Step 3: Simplify the right side of the equation

Now, evaluate the right side of the equation by substituting \(x = 2\). This gives you \(2 + 8\), which simplifies to 10 as well.

Step 4: Compare both sides of the equation

Compare the simplified left side and right side of the equation. Both sides equal 10, so they are the same.

Step 5: Conclusion: Determine if 2 is a solution

Since both sides of the equation are equal when \(x = 2\), the number 2 is indeed a solution to the equation \(4x + 2 = x + 8\).

Key Concepts

Substituting into an equation

When you want to determine if a number is a solution to an equation, the first step is to substitute the given number for the variable in the equation. This is often the variable you're solving for, such as "x." Let's see how it's done!

Imagine we have an equation like \( 4x + 2 = x + 8 \), and we want to check if 2 is a solution. Replace every "x" in the equation with 2, which gives you \( 4(2) + 2 = 2 + 8 \). By doing this, you transform the equation into a plain numerical expression, making it easier to solve.

Substituting is all about swapping the variable with the number in question. From here, we can move onto the next step, which is simplifying.

Simplifying expressions

Once substitution is complete, the next important step is simplifying the expression. Simplification makes expressions easier to handle and helps in getting clear answers.

In our example, after substituting 2 into \( 4x + 2 = x + 8 \), we get \( 4(2) + 2 \) on the left side. Let's break it down:

• First, calculate \( 4 \times 2 \), which equals 8.
• Then, add 2 to that result, giving you 10 for the left side.

For the right side, substitute the same way: \( 2 + 8 \), which also simplifies to 10. Seeing both sides simplified helps to visually confirm equality. Simplification is key to postponing errors and confirming the value substitution.

Comparing sides of an equation

Finally, we come to comparing the sides of the equation. This process determines whether each side has the same value, verifying our solution's correctness.

After simplification, our equation \( 4(2) + 2 = 2 + 8 \) changes to 10 = 10. This tells us that both sides are equal.

In mathematical terms, if the values match, it confirms the number you substituted is a solution to the equation. The beauty of comparing both sides lies in the certainty it provides. Whether solving algebraic equations or real-life problems, ensuring both sides play fair and equal is the key.

• If both sides are equal, the number is indeed a solution.
• If not, the number isn't a solution, and you might need to try another candidate.

With this step, you conclude whether your chosen number makes the equation true.