/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 \(4 z-6=3 z\)... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 40

\(4 z-6=3 z\)

Short Answer

Expert verified
The solution is \( z = 6 \).

Step by step solution

01

- Move All Variable Terms to One Side

To isolate the variable, move all terms involving the variable to one side of the equation. Subtracting 3z from both sides gives: \[ 4z - 3z - 6 = 3z - 3z \] which simplifies to \[ z - 6 = 0 \]
02

- Isolate the Variable

Next, isolate the variable by adding 6 to both sides of the equation: \[ z - 6 + 6 = 0 + 6 \] this simplifies to \[ z = 6 \]
03

- Verify the Solution

Substitute the solution back into the original equation to verify it: \[ 4(6) - 6 = 3(6) \] which simplifies to \[ 24 - 6 = 18 \] Since both sides of the equation are equal, \( z = 6 \) is the correct solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolate the Variable
Isolating the variable is a crucial step in solving linear equations. The goal is to get the variable alone on one side of the equation so you can easily solve for it.
In the exercise provided, we started with the equation: \[ 4z - 6 = 3z \],
To isolate the variable, we need to move all terms involving the variable 'z' to one side of the equation. This was done by subtracting \(3z\) from both sides, yielding:
\[ 4z - 3z - 6 = 3z - 3z \]
\[ z - 6 = 0 \].
This step simplifies the equation and leaves us with only one term containing the variable, making it easier to solve. Remember, the key here is to perform the same operation on both sides of the equation to maintain balance.
Algebraic Manipulation
Algebraic manipulation involves using basic algebraic operations like addition, subtraction, multiplication, and division to simplify and solve equations. After we moved all variable terms to one side, our equation became: \[ z - 6 = 0 \].
The next step is to isolate 'z' completely. This can be achieved by adding 6 to both sides of the equation, which gives us:
\[ z - 6 + 6 = 0 + 6 \]
\[ z = 6 \]
By performing this addition, we have successfully isolated 'z' and determined its value. These manipulations are straightforward but essential for solving more complex equations.
Verify Solution
Verification is the process of substituting the found solution back into the original equation to ensure it holds true. In our equation, we found that \( z = 6 \).
To verify, we substitute \( z = 6 \) back into the original equation \( 4z - 6 = 3z \).
So, the substitution would look like: \[ 4(6) - 6 = 3(6) \]
Simplifying both sides, we get:
\[ 24 - 6 = 18 \]
and
\[ 18 = 18 \]
Since both sides are equal, it confirms that our solution \( z = 6 \) is correct. Always verify your solutions to check for potential errors in your calculations.

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