Problem 17 - College Algebra with Modeling and Visualization

Question

Solve the equation and check your answer. $$ 4(z-8)=z $$

Step-by-Step Solution

Answer

The solution is \( z = \frac{32}{3} \), and it checks out correctly.

Step 1: Distribute the 4

First, we need to distribute the 4 in the expression \( 4(z-8) \). This means we multiply 4 by each term inside the parentheses. This results in \( 4z - 32 = z \).

Step 2: Move Variables to One Side

Next, move all terms containing the variable \( z \) to one side of the equation. Subtract \( z \) from both sides to get \( 4z - z - 32 = 0 \).

Step 3: Simplify the Equation

Simplify the terms by combining like terms. \( 4z - z \) simplifies to \( 3z \) so the equation becomes \( 3z - 32 = 0 \).

Step 4: Isolate the Variable

Add 32 to both sides of the equation to isolate the term with \( z \). This gives us \( 3z = 32 \).

Step 5: Solve for z

Divide both sides by 3 to solve for \( z \):\[ z = \frac{32}{3} \]

Step 6: Check the Solution

Substitute \( z = \frac{32}{3} \) back into the original equation to verify. Calculate \( 4(z - 8) \) as: \[ 4\left(\frac{32}{3} - 8\right) = 4\left(\frac{32 - 24}{3}\right) = 4\left(\frac{8}{3}\right) = \frac{32}{3} \]The left side equals the right side, which confirms our solution is correct.

Key Concepts

Distributive Property

In solving linear equations, understanding the distributive property is pivotal. This property allows us to simplify expressions and solve equations efficiently. In the given exercise, the expression \(4(z-8)\) requires applying the distributive property. Simply put, this involves multiplying the number outside the parentheses by each term inside the parentheses. That means you multiply 4 by both \( z \) and \( -8 \).

• This results in expanding the equation to \( 4z - 32 \).
• Makes it easier to work with the equation and move towards isolating the variable.

Understanding and applying the distributive property helps break down complex expressions into simpler terms, making it easier to manage the rest of the equation solving process.

Isolating Variables

Isolating the variable is a crucial step when solving any equation. This process involves manipulating the equation so that the variable, in this case, \( z \), is alone on one side of the equation. In our equation \(4z - 32 = z\), we start by moving all terms involving \( z \) to one side.

• First, subtract \(z\) from both sides: \(4z - z - 32 = 0\).
• Next, combine the \( z \) terms: \(3z - 32 = 0\).
• Then, add 32 to both sides to help isolate \( 3z \): \(3z = 32\).

Finally, we simplify by dividing each side by 3, leading to the solution \( z = \frac{32}{3} \). Isolating variables lets you find their exact value, as it clarifies how the variable relates to other terms in the equation.

Checking Solutions

Once you have found a solution to the equation, it is important to verify it. This can be done by substituting the solution back into the original equation and checking that both sides are equal. For our solution \( z = \frac{32}{3} \), we substitute this value back into the initial equation \(4(z-8) = z\).

• Compute \(4(z-8)\): \[4\left(\frac{32}{3} - 8\right) = 4\left(\frac{32 - 24}{3}\right) = 4\left(\frac{8}{3}\right) = \frac{32}{3}\]
• Ensure the left side simplifies to match the right side, confirming our solution.

This step ensures the accuracy of your solution, providing confidence that you have correctly solved the equation. Checking solutions is essential to confirm that no mistakes were made during calculations.