Problem 13 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$z+8=-12$$

Step-by-Step Solution

Answer

The solution to the equation \(z+8=-12\) is \(z=-20\).

Step 1: Isolate the Variable

Start solving the equation \(z+8=-12\) by isolating the variable \(z\). To do this, subtract 8 from both sides of the equation. This results in \(z+8-8 = -12-8\). The Addition Property of Equality allows the subtraction of the same number from both sides without violating the balance of the equation.

Step 2: Simplify the Equation

After subtracting 8 from both sides, the equation simplifies to \(z = -20\). Thus, this is the solution to the equation.

Step 3: Check the Solution

It's important to check the proposed solution. Substitute the value of 'z' back into the original equation \(z+8=-12\). Replacing 'z' with '-20' gives \(-20+8=-12\), which simplifies to -12=-12, confirming that the solution is correct.

Key Concepts

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra. A linear equation is an algebraic statement that represents a straight line when graphed on a coordinate plane. These equations typically include one or more variables and are composed of constants and coefficients.
When you encounter an equation like \(z + 8 = -12\), you're dealing with a simple linear equation with one variable, aimed at finding the value of \(z\) that makes the equation true. The strategy for solving such an equation involves isolating the variable on one side of the equation while ensuring that the other side contains the constants. By performing the same operation on both sides—like subtracting 8 in our example—you preserve the equation's balance due to the addition property of equality.
It is not only important to solve for the variable but also to verify your solution. In this case, substituting the value of \(z\) back into the original expression confirms whether you've accurately maintained equality throughout your calculations.

Algebraic Properties

Algebraic properties are rules that allow the manipulation of equations in a way that preserves equality and logical steps. One of these essential properties is the addition property of equality. It states that if you add or subtract the same number from both sides of an equation, the two sides remain equal. This property is crucial when solving equations since it ensures that the operations you perform to isolate a variable do not affect the solution's validity.

Why is this property important?

• It provides a systematic way to manipulate equations.
• It ensures that the equality is maintained throughout the problem-solving process.

The addition property of equality is instrumental not only in solving linear equations but also in other algebraic methods where maintaining equality is essential.

Equation Solving Steps

Understanding the steps to solve an equation is essential for effective problem-solving. The process usually involves a series of logical and systematic steps to isolate the variable and solve for its value. For the equation \(z + 8 = -12\), we see these steps demonstrated clearly:

Step 1: Isolate the Variable

Start by rearranging the equation to bring the variable \(z\) onto one side and the constants onto the other. This often involves adding or subtracting terms from both sides of the equation while applying the addition property of equality.

Step 2: Simplify the Equation

Once the variable is isolated, perform any necessary simplifications such as combining like terms or reducing fractions, if applicable. In our example, subtracting 8 from both sides yielded \(z = -20\).

Step 3: Check the Solution

Always plug the found solution back into the original equation to ensure its correctness. This verification step confidently affirms that the value you have computed is indeed a solution to the equation.