Problem 38 - Introductory Algebra for College Students

Question

Solve each equation in using both the addition and multiplication properties of equality. Check proposed solutions. $$14=5 z-21$$

Step-by-Step Solution

Answer

The solution to the equation 14=5z-21 is z=7.

Step 1: Addition Property of Equality

The first step is to isolate 'z' term on one side. This can be accomplished by adding 21 on both sides of the equation to eliminate the '-21' on the right hand side. This is done as follows: 14 + 21 = 5z - 21 + 21. After performing the addition, our equation becomes: 35 = 5z.

Step 2: Multiplication Property of Equality

Next, to find the value of 'z', divide both sides of the equation by 5. This is done as follows: 35/5 = 5z/5. After performing the division, we get z = 7.

Step 3: Check the Solution

To verify that the proposed solution is correct, substitute 'z' with 7 into the original equation and check if both sides are equal. This is done as follows: 14 = 5*7 - 21. Simplifying the right hand side yields 14 = 14, which show that the solution is correct.

Key Concepts

Addition Property of Equality

The addition property of equality is a fundamental principle that allows us to maintain balance in an equation. When solving for a variable, it's crucial to isolate it.
To do this, safely add or subtract the same number from both sides, ensuring the equation remains equal.
In our example, we wanted to remove \(-21\) from the right side to isolate the variable term. We added 21 to both sides:

• Original: \(14 = 5z - 21\)
• Add 21: \(14 + 21 = 5z - 21 + 21\)
• Simplify: \(35 = 5z\)

This shows how the addition property helps in moving terms across the equation to make solving easier.

Multiplication Property of Equality

The multiplication property of equality lets us solve for a variable by dividing or multiplying both sides of an equation by the same non-zero number.
This helps in getting the variable alone.
In our exercise, once we had \(35 = 5z\), the goal was to solve for \(z\).

• Divide both sides by 5: \(\frac{35}{5} = \frac{5z}{5}\)
• Simplify: \(z = 7\)

Using this property ensures that the equation stays balanced while isolating the variable to find its value.

Checking Solutions

Checking your solution is an important step in solving equations. It helps confirm that you've done everything correctly.
By substituting the result back into the original equation, you can verify accuracy.
Let's check if \(z = 7\) is the correct solution:

• Substitute \(z\) into the original equation: \(14 = 5 \times 7 - 21\)
• Simplify: \(14 = 35 - 21\)
• Result: \(14 = 14\)

Since both sides are equal, our solution is verified and correct!

Introductory Algebra

Introductory algebra introduces basic operations and principles that are essential for solving equations. Understanding properties like addition and multiplication of equality helps form the foundation.
These concepts teach you to manipulate equations while keeping them balanced.
As you solve equations:

• Learn to identify like terms
• Understand the properties of operations
• Develop strategies for isolation of variables

Mastery of these skills is crucial for higher mathematics and daily problem-solving scenarios.