Problem 9 - Prealgebra

Question

Check to see if the number to the right of each of the following equations is the solution to the equation. $$x-4=2 x+1 ;-4$$

Step-by-Step Solution

Answer

No, -4 is not a solution to the equation as both sides are not equal.

Step 1: Examine the Problem

Check to see if the number to the right of each of the following equations is
the solution to the equation.
$$x-4=2 x+1 ;-4$$

Step 2: Apply Mathematical Methods

We use the relevant mathematical definitions and methods to work through the problem.

Step 3: Answer

No, -4 is not a solution to the equation as both sides are not equal.

Key Concepts

Substitution Method

In solving algebraic equations, the substitution method is a powerful tool. This involves replacing a variable with a given number to verify if it satisfies the equation. Let's unravel this step by step using the provided exercise.

In the equation \(x - 4 = 2x + 1\), we are asked to check if \(-4\) is a solution. By substituting \(-4\) for \(x\), we transform the equation into \(-4 - 4 = 2(-4) + 1\). This substitution lets us focus on numbers rather than letters which can make solving more concrete for beginners.

After substitution, ensure that you simplify both sides of the equation separately. Here, the left side becomes \(-8\) and the right side evaluates to \(-7\). Since these values aren't equal, the original equation isn't balanced, indicating that \(-4\) isn't a solution. Using substitution in this way helps in understanding the behavior of variables in equations.

Equation Checking

The process of equation checking is integral to confirming if a proposed solution is correct. Once we've substituted our number into the equation, we need to simplify both sides to verify the equality. In our exercise, after substituting \(-4\), you’d simplify the left side: \(-4 - 4\) which results in \(-8\).

For the right side, \(2(-4) + 1\), calculate \(2(-4)\) which is \(-8\), then add \(1\), giving \(-7\). At this point, compare both sides. Since \(-8eq -7\), the proposed solution does not satisfy the equation.

• Substituting the value into the equation.
• Simplifying each side separately.
• Comparing the results to check if they are equal.

Equation checking provides clarity on whether our values satisfy the given equation or not.

Prealgebra Concepts

Prealgebra is foundational for understanding how algebraic expressions work. Concepts like balancing equations, understanding variables, and performing operations correctly are essential. In our exercise, some key ideas include:

**Balancing Equations:** The equation \(x - 4 = 2x + 1\) suggests a need for balance; both sides must ultimately hold the same value for a given \(x\).

**Variables and Constants:** Recognizing that \(x\) is a placeholder for numbers can be enlightening. Here, we replaced \(x\) with the constant \(-4\) to test the equation.

**Operations:** Simplifying expressions (like \(-4 - 4\) and \(2(-4) + 1\)) helps cement understanding of the order of operations and arithmetic.

By strengthening these prealgebra concepts, students can more confidently approach and solve algebraic equations, leading to a deeper grasp of more complex mathematics in the future.