Problem 21 - Introductory Algebra for College Students

Question

Solve each equation in using the multiplication property of equality. Be sure to check your proposed $$-x=23$$

Step-by-Step Solution

Answer

The value of x that satisfies the equation -x = 23 is x = -23.

Step 1: Multiply both sides by -1

The given equation is -x = 23. In order to change the coefficient of x to a positive value, both sides of the equation are multiplied by -1, yielding: (-1)*(-x) = (-1)*23.

Step 2: Simplify the equation

By multiplication, the equation becomes: x = -23.

Step 3: Verify the solution

Substitute the proposed solution (x=-23) into the original equation to verify. The equation -x = 23 thus becomes: -(-23) = 23, which simplifies to 23 = 23, confirming the solution is valid.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra. It involves finding the value of the variable that makes the equation true. In algebra, we often deal with equations involving variables, such as \(-x = 23\). This type of equation can be solved using the Multiplication Property of Equality.

The key principle behind this property is that you can multiply both sides of the equation by the same non-zero number, and the equation will remain balanced.

Here is how you can solve such an equation step by step:

• Identify the multiplication you need to perform to isolate the variable, here it is \( -x \).
• Apply the multiplication to both sides of the equation to maintain equality. This means multiplying both sides by \(-1\) to get \( x \) on one side.
• Simplify the results to determine the value of the variable, here the equation becomes \( x = -23\).

With these steps, you can efficiently tackle various types of algebraic equations.

Algebraic Equations

Algebraic equations consist of expressions set equal to each other, containing one or more variables. These equations serve as the backbone for algebra.

• Linear equations like \(-x = 23\) involve variables raised to the power of one and are straightforward to solve.
• Equations represent a balance, and the main goal is to keep that balance while solving for the variable.
• Solving involves operations that maintain equality, such as addition, subtraction, multiplication, or division across both sides.

Understanding the type of equation you are dealing with is crucial as it determines which solving techniques or strategies to employ. Simple linear equations, such as the one in our example, often only require basic operations to isolate the variable.

Verification of Solutions

Verifying solutions is an essential step in solving algebraic equations. It ensures that the computed solution is actually correct and satisfies the original equation. Here’s how this process works:

• Substitute the found solution back into the original equation, replacing the variable with the determined value.
• Simplify the equation to see if both sides are equal. If yes, the solution is verified. For example, substituting \( x = -23\) back into \(-x = 23\) yields \(-(-23) = 23\) or \(23 = 23\), confirming the solution is correct.
• Verification provides confidence in the solution and highlights any potential errors made during the calculation process.

Verification is a key practice, especially in exams or problem-solving settings, where accuracy is paramount.