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Chapter 2: Problem 46

Solve the equations. More than one step is needed in each case.$$4-3 x=16$$

Short Answer

Expert verified
x = -4

Step by step solution

01

Isolate the Variable Term

Start by isolating the term with the variable on one side of the equation. To do this, subtract 4 from both sides, which will help move the constant to the other side. This will leave you with:\[-3x = 16 - 4\]Simplifying the right-hand side gives:\[-3x = 12\]
02

Solve for the Variable

Now, solve for \(x\) by dividing both sides of the equation by the coefficient of \(x\), which is -3.\[x = \frac{12}{-3}\]Calculate the division to find \(x\):\[x = -4\]

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Key Concepts

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

Algebra
Algebra is a fundamental branch of mathematics that deals with symbols, often in the form of letters like \(x\) and \(y\), which are used to represent numbers in equations and expressions. It allows us to solve problems where numbers are unknown, making it a versatile tool for various applications.
  • Symbols and Equations: In algebra, symbols stand for numbers, making it easier to generalize mathematical concepts.
  • Algebraic Expressions: These are combinations of variables, numbers, and operations, such as addition or subtraction.
The power of algebra lies in its ability to solve problems by manipulating these symbols to find unknown values. Understanding how to create and solve equations is an essential skill, allowing for the solution of a wide range of mathematical and real-world problems.
Isolating the Variable
Isolating the variable is a crucial step when solving equations. It involves rearranging the equation to get the unknown variable by itself on one side. This step bridges the initial equation to its solution.
  • Understanding Isolation: Think of it as untangling a knot to reach what's important - the variable.
  • Arithmetic Operations: Use basic operations: addition, subtraction, multiplication, or division, to move terms around.
In the example we have, the variable \(x\) was accompanied by numbers. By subtracting a number first and then dividing, we effectively "isolated" \(x\). This step is like peeling off layers to uncover the core of what we need to solve.
Multi-Step Equations
Multi-step equations involve more than one operation to find the solution. They require a series of logical steps to simplify and solve. By breaking these equations down into manageable parts, we develop a systematic approach to reaching the solution.
  • Sequential Steps: Identify and carry out simple operations step-by-step.
  • Order of Operations: Consider the order, often using the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
In the given exercise, the first step was to move the constant by subtracting, followed by isolating the variable by division. Multi-step equations might seem complex, but by tackling each step methodically, they become much more approachable. Remember, patience and practice are keys to mastering these equations.

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Most popular questions from this chapter

Consider a variable \(r\) where \(r\) represents the whole numbers from 1 to 15. Stated mathematically, the possible values of \(r\) are $$r=1,2, \ldots, 14,15$$ Determine which values satisfy the given compound inequalities. $$4 \leq r \leq 8$$

Solve the equations by using the multiplication property. $$\frac{x}{4}=7$$

Solve the equations by using the division property. $$3 x=21$$

Apply the order of operations and answer the questions. The volume of a right rectangular pyramid with a height of 27.6 centimeters and a square base that is 22.1 centimeters on a side is given by \(\frac{22.1^{2} \cdot 27.6}{3} .\) Evaluate the expression and interpret the result. Round the volume to two decimal places.

Use the formula $$ V=\frac{1}{3} \pi\left(\frac{w}{2}\right)^{2} h $$ for the volume \(V\) of a right circular cone, where \(w\) represents the width of the base (this is the same as the diameter) and \(h\) represents the height. Round the results to two decimal places and include proper units. A standard highway traffic cone is 14 inches wide and 30 inches tall. What is the volume of a highway cone?
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