Problem 20 - Introductory Algebra for College Students

Question

Solve each equation in using the multiplication property of equality. Be sure to check your proposed $$-x=17$$

Step-by-Step Solution

Answer

The solution to the equation \(-x = 17\) is \(x = -17\).

Step 1: Multiplication Property of Equality

Multiply each side of the equation \(-x = 17\) by the same number, which is \(-1\). Doing so gives \( (-1) * (-x) = (-1) * 17\).

Step 2: Simplify Both Sides

Simplify both sides of the equation from Step 1, resulting in \( x = -17\).

Step 3: Check the Solution

Substitute \( x = -17\) back into the original equation to verify the solution. On substitution, it gives \(-(-17) = 17\), which verifies as an accurate solution.

Key Concepts

Solving Equations

Solving equations is a fundamental aspect of algebra. It involves finding the value of the unknown variable that makes the equation true. In the context of our original exercise, we are tasked with solving the equation \(-x = 17\), where "\(x\)" is the variable we want to solve for.

To solve equations effectively, we must perform operations that isolate the variable. The goal is to simplify the equation step by step until the unknown variable stands alone on one side of the equation. In this exercise, we achieved this by using algebraic manipulation, specifically through the multiplication property of equality. This guided us to the final solution, which showed that \(x\) equals \(-17\).

The final step in solving equations is to verify our solution. Substituting \(-17\) back into the original equation, \(-(-17) = 17\), confirms our solution is accurate, as both sides of the equation equal the same value. This verification step is crucial to ensure no mistakes were made during the solving process.

Properties of Equality

The properties of equality are rules that allow us to balance and manipulate equations. These rules ensure that equations stay equivalent as we solve for unknowns. There are several key properties, and the one used in the original solution is the multiplication property of equality.

• The multiplication property of equality states that if you multiply both sides of an equation by the same non-zero number, the two sides remain equal.
• For the equation \(-x = 17\), we applied this property by multiplying both sides by \(-1\) to isolate the variable \(x\).

Understanding and correctly applying the properties of equality are essential for solving equations accurately. They provide the foundation that allows algebraic manipulation to maintain the truth of an equation throughout the solving process.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to transform equations into simpler, solvable forms. It is an essential skill for solving equations and involves techniques like adding, subtracting, multiplying, or dividing both sides of an equation by the same number.

In the provided exercise, we utilized multiplication as our tool for algebraic manipulation. By multiplying both sides of \(-x = 17\) by \(-1\), we transform the equation into \(x = -17\). This process of changing the equation without altering its truth is what makes algebraic manipulation so critical.

• It allows for the simplification of complex equations into easily solvable ones.
• It applies the principles of equality to uphold the validity of the equation.
• Skills in algebraic manipulation are foundational for advancing in algebra and other mathematical disciplines.

By mastering these skills, you can tackle a wide range of algebraic challenges with confidence.