Problem 18 - College Algebra with Modeling and Visualization

Question

Solve the equation and check your answer. $$ -3(2 z-1)=2 z $$

Step-by-Step Solution

Answer

The solution to the equation is \(z = \frac{3}{8}\).

Step 1: Distribute the -3

First, distribute the -3 across the terms inside the parentheses: \(-3(2z - 1)\). This results in \(-6z + 3\). Now, the equation is: \[-6z + 3 = 2z\].

Step 2: Move all terms involving z to one side

To isolate terms involving \(z\), add \(6z\) to both sides of the equation: \[-6z + 3 + 6z = 2z + 6z\]. Simplifying gives: \[3 = 8z\].

Step 3: Solve for z

To solve for \(z\), divide both sides of the equation by 8: \[\frac{3}{8} = z\]. Thus, \(z = \frac{3}{8}\).

Step 4: Check the solution

Substitute \(z = \frac{3}{8}\) back into the original equation to verify the solution: \[-3(2(\frac{3}{8})-1) = 2(\frac{3}{8})\]. Calculate both sides: \[-3(\frac{6}{8} - 1) = \frac{6}{8}\]. Simplifying the expression: \[-3(-\frac{1}{4}) = \frac{3}{4}\] \[\frac{3}{4} = \frac{3}{4}\].Since both sides of the equation are equal, the solution is verified.

Key Concepts

Distributive Property

The distributive property is a very useful tool in algebra that makes it easier to simplify expressions and solve equations. It allows us to remove parentheses by multiplying each term inside by the number outside. If you have an expression like \( a(b + c) \), the distributive property tells us that you can transform it into \( ab + ac \). This step is crucial when you are faced with equations that include parentheses.

In the original exercise, we had the equation:

• \(-3(2z - 1) = 2z\)

We used the distributive property to expand the left side of the equation. By multiplying \(-3\) with both \(2z\) and \(-1\), we got:

• \(-6z + 3 = 2z\)

This step made it possible to isolate the variables later on. Whenever you see parentheses in an algebraic expression or equation, remember that the distributive property can help you simplify your work and make the math more straightforward.

Isolating Variables

Isolating variables is an essential step when solving algebraic equations. The goal is to get the variable you are solving for by itself on one side of the equation.

After using the distributive property in the original problem, the equation became:

• \(-6z + 3 = 2z\)

To isolate the variable \(z\), we aim to move all terms involving \(z\) to one side of the equation. This is done by adding or subtracting terms.

For the equation above, we added \(6z\) to both sides to get:

• \(3 = 8z\)

With this simpler equation, the next step is straightforward. Divide both sides by 8 to solve for \(z\). This gives us:

• \(z = \frac{3}{8}\)

This technique, isolating the variable, is widely used in algebra and crucial for equation solving. By isolating the variable, you’re essentially narrowing down the equation to its simplest form.

Equation Checking

Once you've found your solution, it’s important to check your work to ensure it’s correct. This last step involves substituting your solution back into the original equation.

From the exercise, we found \(z = \frac{3}{8}\). To check, we substitute \(\frac{3}{8}\) back into the original equation \(-3(2z - 1) = 2z\):

• Calculate the left side: \(-3(2(\frac{3}{8}) - 1)\).
• Calculate the right side: \(2(\frac{3}{8})\).

Simplifying the expression, both sides turned out to be equal:

• \(-3(-\frac{1}{4}) = \frac{3}{4}\)

If both sides of the equation are equal after substitution, the solution is verified. This step not only confirms the solution but also builds understanding and confidence. It proves whether or not the solution satisfies the original equation.