Problem 10 - Introductory Algebra for College Students

Question

Solve each equation in using the multiplication property of equality. Be sure to check your proposed $$-18=-3 z$$

Step-by-Step Solution

Answer

In the equation \(-18=-3z\), the solution for the unknown variable \(z\) is 6.

Step 1: Identify the coefficient of the unknown

In the equation \(-18=-3z\), the unknown variable is \(z\) and it has a coefficient of \(-3\).

Step 2: Use the multiplication property of equality to solve for the unknown

To isolate \(z\), divide both sides of the equation by \(-3\). Doing so, the equation becomes \(-18/(-3) = -3z/(-3)\). This simplifies to \(6=z\).

Step 3: Check the solution

Substitute \(z = 6\) back into the original equation to check. \(-18 = -3 \cdot 6\). This simplifies to \(-18 = -18\). Since both sides of the equation are same, the solution \(z = 6\) is verified.

Key Concepts

Solving Equations

Solving equations is a foundational skill in algebra used to find the value of an unknown variable. Equations are like puzzles, where you need to find the missing piece that makes the whole picture complete. In the exercise provided, we had the equation \(-18 = -3z\). Our task was to determine the value of \(z\) that makes this equation true. The steps to solve it involve identifying what we need to isolate (the variable) and using mathematical operations to achieve that isolation.

The basis of solving equations is to perform the same operation on both sides of the equation. This is the essence of equality: whatever you do to one side, you must do to the other. The ultimate goal is to have the variable by itself on one side of the equation to clearly see what it equals.

Here, the goal was to isolate \(z\), which we did by dividing both sides of the equation by \(-3\), using the multiplication property of equality.

Algebraic Manipulation

Algebraic manipulation involves rearranging an equation to better understand or solve it. It’s like untying knots to straighten out a string. In the context of our equation, \(-18 = -3z\), the algebraic manipulation was quite straightforward. The variable \(z\) was being multiplied by \(-3\), so we reversed this by performing the opposite operation: division.

The multiplication property of equality tells us that if we multiply or divide both sides of an equation by the same non-zero number, the sides remain equal. For example:

• Original equation: \(-18 = -3z\)
• Dividing both sides by \(-3\): \(-18/-3 = -3z/-3\)
• Resulting in: \(6 = z\)

This handy property helps maintain the balance of the equation while we isolate the variable. Remember, the aim is to make the variable clear on one side to easily identify its value.

Checking Solutions

Once we propose a solution to an equation, it's crucial to check if it actually works. Checking solutions confirms that your answer is correct. Like double-checking your work to ensure accuracy, it involves substituting the solution back into the original equation to see if it holds true.

In our case, the proposed solution was \(z = 6\). To check it, we substitute \(z\) back into the original equation:

• Substitute \(z = 6\) into \(-18 = -3z\)
• You get: \(-18 = -3 \cdot 6\)
• Simplifying the right-hand side gives \(-18 = -18\)

Since both sides of the equation are equal, our solution \(z = 6\) is verified. This step is essential as a dummy check to ensure all operations and steps were correct and led to a true statement. Without this confirmation, there's a risk of having made an error unnoticed.