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Chapter 2: Problem 46

\(\frac{2}{7} z-2=\frac{9}{7} z\)

Short Answer

Expert verified
z = -2

Step by step solution

01

Isolate variable terms

Start by moving all terms involving the variable z to one side of the equation. Subtract \(\frac{2}{7}z\) from both sides: \[ \frac{2}{7}z - 2 - \frac{2}{7}z = \frac{9}{7}z - \frac{2}{7}z \]
02

Simplify the equation

Combine like terms. On the left side, \(\frac{2}{7}z - \frac{2}{7}z = 0\), and on the right side, \(\frac{9}{7}z - \frac{2}{7}z = \frac{7}{7}z = z\). This simplifies the equation to: \[ -2 = z \]
03

Write the final solution

The variable z is now isolated and its value can be written as: \[ z = -2 \]

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Key Concepts

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

isolating variables
To solve a linear equation like \(\frac{2}{7} z-2=\frac{9}{7} z\), we first need to isolate the variable we're solving for, which in this case is \(z\). Isolating a variable means getting it by itself on one side of the equation. This involves:
  • Identifying terms that contain the variable.
  • Moving all other terms to the opposite side of the equation.
For our equation, the steps are as follows:
  • First, identify the variable terms, \(\frac{2}{7}z\) and \(\frac{9}{7}z\).
  • Subtract \(\frac{2}{7}z\) from both sides to start isolating z: \(\frac{2}{7}z - 2 - \frac{2}{7}z = \frac{9}{7}z - \frac{2}{7}z\).
This simplification brings us closer to isolating \(z\) by itself on one side.
combining like terms
After we've moved terms around to isolate the variable, our next task is often to combine like terms. Like terms are terms that contain the same variable raised to the same power. Combining them simplifies the equation.
In our example, we need to simplify both sides after isolating \(z\):
\(\frac{2}{7}z - \frac{2}{7}z = 0\) on the left and \(\frac{9}{7}z - \frac{2}{7}z = \frac{7}{7}z = z\) on the right.
Notice how the terms with \(z\) are combined to make our equation simpler:
\(-2 = z\).
Simplifying equations by combining like terms reduces the complexity, making the solution more straightforward.
rational coefficients
Rational coefficients are coefficients that are fractions. For instance, \(\frac{2}{7}\) and \(\frac{9}{7}\) in our original equation \(\frac{2}{7} z-2=\frac{9}{7} z\). When working with rational coefficients:
  • Perform operations like addition, subtraction, and multiplication as you would with whole numbers but keep the fractions in mind.
  • Simplify fractions wherever possible to make the equation easier to work with.
Here, both \(\frac{2}{7}z\) and \(\frac{9}{7}z\) are fractions, and we need to carefully handle these coefficients while performing operations:
  • Subtracting \(\frac{2}{7}z\) from \(\frac{9}{7}z\) to get \(\frac{7}{7}z = z\).
Understanding how to manipulate rational coefficients properly is crucial for solving linear equations accurately.

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