Problem 20 - Introductory Algebra for College Students

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 to the equation \(18 + z = 14\) is \(z = -4\)

Step 1: Subtract 18 from both sides

Use the addition property of equality to subtract 18 from both sides of the equation. This gives: \(z = 14 - 18\)

Step 2: Simplify

Simplify the equation on the right-hand side to find the value of z. This gives: \(z = -4\)

Step 3: Check the solution

Substitute the computed value of z into the original equation to see if it holds true. Substituting z = -4 into the original equation yields \(18 - 4 = 14\), which is a true statement. This confirms that the solution is correct.

Key Concepts

Solving Equations

Equations are mathematical statements that show equality, meaning that the expressions on both sides of the equation symbol, '=', are equal. Solving equations involves finding the value of the variable that makes the equation true. Specifically, in the context of this exercise, we use a method known as the **addition property of equality**. This property allows us to add or subtract the same number from both sides of an equation without changing the equality. This property helps isolate the variable on one side of the equation.

• Start by identifying operations on the variable.
• Use the addition property of equality to remove those operations.
• Subtract or add values to both sides to isolate the variable.

In our example `18 + z = 14`, by subtracting 18 from both sides, we simplify our target equation, leading us to the desired variable value.

Algebraic Equations

Algebraic equations consist of variables and numbers combined through operations like addition, subtraction, multiplication, or division. Understanding algebraic equations is crucial because they form the foundation of algebra. In the equation `18 + z = 14`, 'z' is our variable, and we're interested in determining its value.
Using addition and subtraction can help simplify equations and find solutions. This is because it keeps the equation balanced, or "equal," throughout the solving process. Any addition or subtraction performed on one side of the equation must also be performed on the other side to maintain equality.
Here's a simple breakdown:

• Identify the variable and its accompanying number.
• Use operations to find the variable on one side of the equation.
• Simplify values to solve the equation efficiently.

Check Solutions

Once you've found a potential solution to an equation, it's important to verify it to ensure accuracy. This step ensures that the solution we identified satisfies the original equation, confirming correctness. Here's how to check solutions effectively:

• Substitute the solution back into the original equation.
• Perform calculations to see if both sides of the equation are balanced.
• If both sides are equal, your solution is correct.

For instance, in our original problem, after identifying `z = -4`, substituting `-4` back into the equation `18 + z` should yield `14`. By confirming that substituting `z = -4` results in `18 - 4 = 14`, we ascertain our solution is accurate.