Problem 12 - Fundamentals of Mathematics

Question

Verify that each given value is a solution to the given equation. $$x-11=5, x=16$$

Step-by-Step Solution

Answer

Yes, \( x = 16 \) is a solution to the equation \( x - 11 = 5 \).

Step 1: Understand the equation

The equation given is \( x - 11 = 5 \). This means that when something is subtracted from \( x \), it results in 5. Here, you have to verify if the given value of \( x \), which is 16, satisfies this equation.

Step 2: Substitute the value into the equation

Substitute 16 for \( x \) in the equation: \( x - 11 = 5 \). This becomes \( 16 - 11 = 5 \).

Step 3: Simplify the equation

Perform the subtraction: \( 16 - 11 = 5 \). This simplifies to \( 5 = 5 \).

Step 4: Verify the result

Since \( 5 = 5 \) is a true statement, the calculation confirms that when \( x = 16 \), the equation \( x - 11 = 5 \) holds true.

Key Concepts

Algebra Basics

Algebra is a branch of mathematics that uses symbols to represent numbers and express mathematical relationships. These symbols, often represented by letters like \( x \), \( y \), and \( z \), make it possible to generalize arithmetic operations. In algebra, equations are used to show the equality between two expressions. An equation consists of two sides, typically separated by an equal sign \( = \). It is essential to remember that both sides of the equation must remain equal to maintain its validity.

Understanding algebra allows you to solve a variety of mathematical problems by performing operations like addition, subtraction, multiplication, and division on the symbols to arrive at the desired value. This skill is highly beneficial, not just in academics but also in day-to-day problem-solving.

Substitution Method for Solving Equations

The substitution method is a straightforward approach to verifying solutions of equations in algebra. It involves replacing the variable in the equation with a specific numeric value. This replacement incorporates the given value into the equation to determine if it satisfies the equation's conditions.

• First, identify the variable you need to substitute. In our example, the equation was \( x - 11 = 5 \) with \( x = 16 \).
• Next, replace \( x \) with 16, so it becomes \( 16 - 11 \).
• Then solve the new expression, which leads to checking if both sides of the equation are equal.

The substitution method helps simplify complex equations by breaking them down into more manageable portions. It is particularly useful in verifying that a specific solution is correct.

Solution Check: Confirming the Answer

After substituting the given values and simplifying the equation, the next crucial step is to perform a solution check. This involves comparing both sides of the equation to ensure that they are equivalent.

In our example, after substituting \( x = 16 \) into the equation \( x - 11 = 5 \), we simplified it to \( 16 - 11 = 5 \). Simplifying this gave us \( 5 = 5 \). Because both sides are equal, the solution is confirmed to be correct.

• Always perform a solution check to validate that a substitution has led to a true equation.
• It ensures the correctness of your solution and the steps that you've followed.
• Even small arithmetic errors can mislead, so double-check your work.

The process of checking solutions is not merely about arriving at the right answer but also about understanding and confirming each step taken during the problem-solving process.