Problem 44 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether the given number is a solution of the equation. $$x+17=22 ; 5$$

Step-by-Step Solution

Answer

Yes, 5 is indeed a solution of the equation \(x+17=22\).

Step 1: Substitution

Start by substituting the given number into the equation in the place of the variable. The variable here is \(x\), and the given number is 5. Therefore, the equation \(x+17=22\) becomes \(5+17=22\).

Step 2: Simplification

Next, simplify the left-hand side of the equation to see if it equals the right-hand side. On simplifying, \(5+17\) equals 22.

Step 3: Verification

On simplifying, the left-hand side equals 22, which is the same as the right-hand side. Therefore, the equation holds, proving that the given number, 5, is indeed a solution of the equation.

Key Concepts

Substitution Method

The substitution method is a fundamental technique for solving equations. It involves replacing a variable with a given number to test if it makes the equation true. In our exercise, we have the equation \(x+17=22\) and need to verify if 5 is a solution.

• The variable \(x\) is our unknown, which we substitute with the number 5.
• This transforms the equation into \(5+17=22\), where 5 replaces \(x\).

This method is quite straightforward. You simply plug in the given number for the unknown variable. It's akin to solving a real-world puzzle where you check if the pieces fit together perfectly.

Equation Simplification

Simplifying an equation means reducing it to its simplest form. This helps clarify whether the left side equals the right side. In our example, \(5+17=22\) is already a simplified version of the original equation. Simplification often involves basic arithmetic.

• Adding \(5\) and \(17\) gives us \(22\).
• Thus, \(5+17=22\) indeed simplifies to \(22=22\).

This step is crucial because it confirms or denies any initial assumptions we made when substituting. A balanced equation ultimately means that the given number does satisfy the equation. If the expressions on both sides match after simplification, your work confirms the precision of your solution.

Solution Verification

Verification is the final step in solving equations. It involves checking that the simplified equation holds true throughout. Here, we initially substituted \(x\) with 5 and simplified to \(22=22\). Verification ensures that not only do these calculations add up, but they also remain consistent.

• If the left side equals the right after simplification, the solution is verified.
• In this case, since \(22\) equals \(22\), our substitution is correct.

Verification confirms the validity of the solution, ensuring that every step follows logically and accurately from the previous one. It adds a layer of certainty and confidence to solve equations correctly.