Problem 40 - Introductory Algebra for College Students

Question

Solve each equation in using both the addition and multiplication properties of equality. Check proposed solutions. $$-x-5=5$$

Step-by-Step Solution

Answer

-10

Step 1: Add 5 to both sides of the equation

Utilize the addition property of equality by adding 5 to each side of the given equation. This will prevent any changes in the solution while isolating the \(x\) term. So, the equation \(-x-5=5\) becomes \(-x=10\).

Step 2: Multiply both sides of the equation by -1

Now it's time to deploy the multiplication property of equality in order to find the value for \(x\). Multiply both sides of the equation by -1 to derive \(x= -10\).

Step 3: Check the solution

Substitute \(x= -10\) into the original equation to check if it satisfies the given equation. Therefore, when substituted, \(-(-10)-5=5\) simplifies to \(10-5=5\) which holds true. Therefore, the solution \(x=-10\) is validated.

Key Concepts

Addition Property of Equality

The addition property of equality is a foundational concept in algebra that helps us solve equations effectively. Imagine you're balancing scales with weights on each side. If you add the same weight to both sides, the scales remain balanced, don't they? Similarly, this property tells us we can add the same number to both sides of an equation without changing the equation’s balance or equality.

For example, if we start with the equation \(-x - 5 = 5\), and we add 5 to each side, what we are doing is keeping things fair and equal. This transformation can be shown as:

• Original equation: \(-x - 5 = 5\)
• Addition of 5 on both sides: \(-x - 5 + 5 = 5 + 5\)
• Resulting equation: \(-x = 10\)

This method doesn’t change the validity of the equation but simplifies our job to isolate the variable "\(x\)." Once we have simplified it to a point where the variable is more manageable, we can further employ other algebraic properties to solve for "\(x\)."

Multiplication Property of Equality

Once you've utilized addition to simplify an equation, you may find the multiplication property of equality handy for solving it entirely. Imagine a seesaw. If every person (representing amounts or numbers) has exactly twice their weight on both sides, the seesaw remains balanced. Similarly, multiplying both sides of an equation by the same non-zero value preserves the equation's balance.

Continuing from the equation \(-x = 10\), we use the multiplication property by multiplying both sides by \(-1\) to make "\(x\)" positive.

• Equation after addition: \(-x = 10\)
• Multiplying both sides by \(-1\): \(-1 \cdot (-x) = -1 \cdot 10\)
• Simplified form: \(x = -10\)

By maintaining the equality of the equation through identical multiplicative actions on both sides, we ensure that our solution, \(x = -10\), is accurate and valid. Each step of multiplication aids in unearthing the true potential solution for the variable at hand.

Algebraic Equations

Algebraic equations are like puzzles waiting to be solved. They contain variables, numbers, and often operation symbols like addition, subtraction, multiplication, or division that guide us toward a solution. These equations are essentially statements of equality demonstrating how two expressions relate to each other.

Consider the initial equation from our exercise, \(-x - 5 = 5\). This algebraic equation serves as a statement implying one expression, \(-x - 5\), is equal to the number \(5\). The goal in solving an algebraic equation is to find the value of the unknown variable, in this case, "\(x\)," that makes this statement true.

To successfully navigate through algebraic equations, remember these steps:

• Identify the variable you need to solve for.
• Use algebraic properties, such as addition or multiplication of equality, to isolate the variable.
• Simplify step by step, ensuring each action preserves the equality.
• Check your final solution by substituting it back into the original equation.

These equations not only help in academic exercises but also enable us to model and solve real-life problems, assisting with anything from calculation-related decisions to scientific data analysis. Understanding the process of solving algebraic equations offers you a powerful tool in mathematics and beyond.