Problem 12 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$y-5=-18$$

Step-by-Step Solution

Answer

The solution for the equation \(y - 5 = -18\) is \(y = -13\).

Step 1: Apply Addition Property of Equality

By applying the addition property of equality, add +5 to both side of the given equation \(y - 5 = -18\). Doing so will get the equation like \(y - 5 + 5 = -18 + 5\).

Step 2: Simplify the Equation

Simplify the equation by performing the operations. It gives a new equation \(y = -18 + 5\). Now simplify the right side by adding -18 and 5, which results in \(y = -13\).

Step 3: Check the Solution

Check the proposed solution \(y = -13\) by substituting -13 for y in the original equation \(y - 5 = -18\). The left side of the equation \(y - 5\) becomes \(-13 - 5\) which simplifies to \(-18\). As the right side of the original equation is also -18, the original equation \(y - 5 = -18\) is satisfied.

Key Concepts

Solving Linear Equations

Understanding how to solve linear equations is essential in algebra. A linear equation is made up of two expressions set equal to each other, with an unknown variable usually represented by letters like 'x' or 'y'. The goal is to isolate the variable on one side of the equation to find its value. In the example we're looking at, the equation to solve is \(y - 5 = -18\).

To solve linear equations, we use various properties, but one fundamental technique is to perform the same operation on both sides of the equation to maintain the balance—a concept stemming from the addition property of equality. This property states that you can add the same number to both sides of an equation without changing the equation's solution. In this exercise, adding 5 to both sides adheres to this rule, which effectively cancels out the '-5' on the left and gets us closer to finding the value of 'y'.

Algebraic Properties

Algebraic properties are the rules that govern the manipulation of equations and enable us to solve them. Among these properties, the addition property of equality is particularly significant because it allows us to change the form of an equation while keeping the solutions the same. It ensures that whatever you do to one side of the equation, doing the same to the other side will not affect the equation's balance.

Apart from the addition property, other algebraic properties include the multiplication property of equality, distributive property, commutative and associative properties. Each of these has a specific role, whether it's distributing a value across a parenthesis, changing the order in which you add or multiply, or grouping numbers in a different way without affecting their sum or product.

Equation Solving Steps

To effectively solve equations, it's helpful to follow a structured approach. The steps taken in the given exercise provide a clear template:

• Step 1: Apply the relevant algebraic property. In our exercise, this meant using the addition property of equality to add 5 to both sides of the equation \(y - 5 = -18\), yielding \(y - 5 + 5 = -18 + 5\).
• Step 2: Simplify the equation. After adding 5 to both sides, the equation simplified to \(y = -13\), making it clearly visible what 'y' is equal to.
• Step 3: Verify the solution by substituting the value back into the original equation to ensure that it makes both sides equal.

Consistently following these steps can significantly streamline the process of solving linear equations and helps to avoid common mistakes.

Verifying Solutions

Once you've arrived at a proposed solution for an equation, it's important to make sure that it's correct by verifying it. This involves substituting the solution back into the original equation and ensuring that it holds true.

For the exercise \(y - 5 = -18\), we found the solution \(y = -13\). To check it, we replace 'y' with -13 in the original equation to get \(-13 - 5\), which simplifies to \(-18\). This matches the right side of the equation, confirming that -13 is indeed the correct solution. This step is crucial as it confirms the validity of the answer and helps in understanding the relationship between the equations and their solutions.