Problem 5 - Prealgebra

Question

Check to see if the number to the right of each of the following equations is the solution to the equation. $$2 x-4=2 ; 4$$

Step-by-Step Solution

Answer

No, \(x = 4\) is not the solution to the equation.

Step 1: Understand the Equation

We are given the equation \(2x - 4 = 2\) and we need to check if \(x = 4\) is the solution. This means we will substitute \(x\) with 4 in the equation and see if the equation holds true.

Step 2: Substitute x with 4

Replace \(x\) with 4 in the equation. This gives us \(2(4) - 4 = 2\).

Step 3: Simplify the Equation

Calculate \(2 \times 4\), which equals 8. Then the equation becomes \(8 - 4 = 2\).

Step 4: Check the Solution

Subtract 4 from 8. This results in 4, so the equation simplifies to \(4 = 2\). Since 4 does not equal 2, substitution of \(x = 4\) does not satisfy the equation.

Key Concepts

Substitution Method

The substitution method is a straightforward technique used in algebra to evaluate whether a specific number is a solution to an equation. This process involves replacing the variable in the equation with the given number and simplifying to verify if the equation is true. In our exercise, the equation is \(2x - 4 = 2\), and the task is to check if \(x = 4\) satisfies this equation.

• Substitute 4 for \(x\): Begin by inserting the number 4 in place of \(x\) in the equation, transforming it into \(2(4) - 4 = 2\).
• Simplify: Perform the operations to simplify the equation and see if both sides are equal.

Using substitution is a valuable skill in algebra, as it allows one to test potential solutions methodically.

Basic Algebra

Basic algebra is the foundation on which higher-level algebraic concepts are built. It involves manipulating equations and expressions using arithmetic and algebraic operations. In our example equation \(2x - 4 = 2\), we perform the following key operations:

• Multiplication: Compute \(2 \times 4\), which results in 8. This is an essential step in handling equations involving coefficients and variables.
• Subtraction: Next, subtract 4 from 8. This simplifies the equation further, bringing it down to \(8 - 4 = 2\).

Each step requires attention to ensure accuracy and proper simplification. Basic algebra skills are crucial when solving equations, as they provide the tools to manipulate and transform equations efficiently.

Solution Verification

Solution verification is the practice of confirming whether a proposed solution actually satisfies an equation. In basic algebra, this often involves substituting a value back into the original equation. In our problem, after simplification, the equation becomes \(4 = 2\). Here's how solution verification works:

• Verify Equality: Once substituted and simplified, check to see if both sides of the equation are equal.
• Assessment: If they are not equal, as in our example where 4 does not equal 2, this indicates the number is not a solution.

Solution verification is crucial because it ensures the correctness of the proposed solution. It's not enough to find a value that looks right; it must be tested and proven, confirming the solution's validity.