Problem 20 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation. Using the addition property of equality. Be sure to check your proposed solutions. $$18+z=14$$

Step-by-Step Solution

Answer

The solution is \(z = -4\).

Step 1: Subtract 18 from both sides

The equation is \(18+z=14\). To isolate \(z\), subtract 18 from both sides of the equation. Doing this gives: \(z = 14 - 18\).

Step 2: Simplify the right side

Subtracting 18 from 14 gives a value of -4. Therefore, the equation simplifies to: \(z = -4\).

Step 3: Check the solution

To check if \(z = -4\) is the correct solution, substitute -4 for \(z\) in the original equation: \(18 + (-4) = 14\). Simplifying this does yield 14, confirming that \(z = -4\) is the correct solution.

Key Concepts

Solving Equations

Solving an equation means finding the value of the unknown variable that makes the equation true. In this process, you use various mathematical operations to simplify the equation and unravel the mystery number.

Whenever you're tasked with solving equations, parameters might vary, but the end goal remains the same: to find the value that satisfies the equation. With the equation \(18 + z = 14\), solving it requires the application of the addition property of equality. This property states that you can add or subtract the same number from both sides of the equation without changing the equation's truth.

By applying this property and simplifying both sides of the equation, you can reveal the value of \(z\) and turn a complex puzzle into a solvable problem!

Isolation of Variable

Isolation of the variable is a key step in solving equations. It involves manipulating the equation in such a way that the variable stands alone on one side. Think of it like peeling an onion to get to the core; you strip away layers until only the variable is left.

The main operations to isolate a variable include:

• Addition or subtraction to remove any constants from the variable's side.
• Division or multiplication to handle coefficients associated with the variable.

In our example, the equation is \(18 + z = 14\). To isolate \(z\), you subtract 18 from both sides to eliminate the constant next to \(z\). This results in \(z = 14 - 18\).

Simple arithmetic will then give you the final isolated variable, showing the value of \(z\) as -4. The goal is clarity — leaving no room for doubt about what the variable equals.

Checking Solutions

After finding a possible solution, it's crucial to check its validity. Just like double-checking your work on a test, confirming your solution ensures accuracy.

To check your solution:

• Substitute the found value back into the original equation.
• Simplify both sides to see if they match.

Let’s apply this to the equation \(18 + z = 14\) with \(z = -4\). Substitute \(-4\) for \(z\): \[18 + (-4) = 14\]This simplifies to \(14 = 14\), proving that our calculation was correct.

Checking solutions is like the last step of a recipe; it confirms that all steps before were done correctly, giving you the confidence that the solution truly satisfies the original equation.