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Chapter 4: Problem 69

Solve the equation if possible. $$ 9-5 z=-8 z $$

Short Answer

Expert verified
The solution to the equation is \( z = 3 \).

Step by step solution

01

Rearranging the equation

Start by moving all terms containing \( z \) to one side of the equation and remaining terms to the other side. This can be done by adding \( 5z \) to both sides of the equation. The equation then becomes: \( 9 = 3z \).
02

Solving for \( z \)

Finally, to solve for \( z \), divide both sides of the equation by 3. This gives: \( z = \frac{9}{3} \).

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Key Concepts

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

Solving Equations
Solving a linear equation involves finding the value of the variable that makes the equation true. Linear equations typically involve terms with variables raised to the first power, constants, and operations like addition, subtraction, multiplication, or division.

To tackle linear equations:
  • Look for like terms, which are terms that contain the same variable raised to the same power.
  • Move all terms involving the variable to one side of the equation by using addition or subtraction.
  • Perform algebraic operations to isolate the variable on one side, solving for its value.

A step-by-step process involves manipulating the equation logically, so at each stage, the equation remains balanced until the variable is isolated on one side.
Variables
Variables are symbols, often represented by letters like \( z \), that stand for unknown values in mathematical expressions and equations. They are essential tools in algebra because they allow for generalization and abstraction. In the equation \( 9 - 5z = -8z \), the variable \( z \) represents the unknown number we aim to find.

Understanding variables involves:
  • Identifying them within the equation.
  • Realizing that they can take on various values, which we determine through solving.
  • Appreciating their role as placeholders that can simplify complex real-world problems into manageable algebraic expressions.
Variables allow expressions to be transformed and solved, enabling us to find specific solutions or even a range of solutions.
Algebraic Manipulation
Algebraic manipulation is the process of rearranging and simplifying equations or expressions to find unknown values. This involves techniques such as:
  • Adding or subtracting the same number or term from both sides of an equation to maintain balance.
  • Combining like terms, which makes it simpler to isolate the variable.
  • Using multiplication or division to solve for the variable when it is a coefficient in the equation.
For instance, in the given equation \( 9 - 5z = -8z \), algebraic manipulation allows us to move \( 5z \) to the other side, turning it into \( 9 = 3z \).

Such manipulations are crucial because they provide the pathway to simplify complex problems systematically, ensuring that the relationships between variables and constants are preserved.

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Most popular questions from this chapter

LOGICAL REASONING In Exercises \(56-59\), tell whether the statement is true or false. Justify your answer. The point \((3,5)\) is on the graph of \(3 x+5 y=30\)

They can be used to estimate the average number of hours \(h\) per week that a household in the United States watches television programs. In all three models, \(t\) is the number of years since \(1985 .\) Hours spent watching television: \(h=0.57 t+54.85\) Hours spent watching talk shows: \(\quad h=0.35 t+4.06\) Hours spent watching game shows: \(h=-0.2 t+3.4\) According to the model, in what year will the average number of hours per week that a household watches game shows be zero?

Find the difference. $$ -8-9 $$

Decide whether the given point lies on the line. Justify your answer both algebraically and graphically. $$-4 x-3 y=-8 ;(-4,2)$$

Decide whether the given point lies on the line. Justify your answer both algebraically and graphically. $$x-y=10 ;(5,-5)$$
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