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Chapter 1: Problem 12

Mentally solve each equation. $$ 6 x=-24 $$

Short Answer

Expert verified
x = -4

Step by step solution

01

Isolate the Variable

To isolate the variable, you need to get rid of the coefficient of the variable. In this case, the variable is \(x\) and the coefficient is 6. To do this, divide both sides of the equation by the coefficient of \(x\).
02

Simplify Both Sides

Divide both sides of the equation by 6:$$\frac{6x}{6} = \frac{-24}{6}$$This simplifies to:$$x = -4$$

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Key Concepts

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

Isolating Variables
When solving any equation, a key step is to isolate the variable. The variable is the unknown quantity that we are trying to find. For example, in the equation given, the variable is \(x\). The goal here is to make \(x\) stand on its own on one side of the equation.
To isolate \(x\), we perform operations that cancel out other elements around it, like coefficients or constants. These operations should always be done on both sides of the equation to keep it balanced.
Essentially, think of yourself as a detective: you want to get the variable by itself so you can 'solve the case' and find its value.
Coefficients
In the equation \(6x = -24\), the number 6 is known as the coefficient. Coefficients are numbers that are directly multiplied by the variable. They play a significant role in algebra as they show how many times the variable is taken.
To remove the coefficient and isolate the variable, you need to do the inverse operation. In this case, 6 is multiplying \(x\), so the inverse operation is division. Therefore, you divide both sides of the equation by the coefficient (6) to isolate \(x\).
When you do this, you鈥檒l end up with the isolated variable: \(x = -4\).
Simplifying Equations
Simplifying an equation means making it as straightforward as possible. After isolating the variable, the next step is to simplify any operations. In our example: \[6x = -24\]
After dividing both sides by 6, we simplify as follows: \[x = \frac{6x}{6} = \frac{-24}{6}\]
This gives us: \(x = -4\).
Simplifying equations often involves performing basic arithmetic operations like addition, subtraction, multiplication, or division. The goal is to reduce the equation to its simplest form, making it easier to understand and solve.
Basic Algebra
Algebra is all about finding the unknowns. When you鈥檙e solving linear equations, you鈥檙e typically working with basic algebra. This means you鈥檙e dealing with simple equations that involve basic mathematical operations.
Linear equations like \(6x = -24\) are foundational to algebra. They involve variables, coefficients, and constants. Solving these equations helps you understand more complex problems later on.
Basic algebra teaches you important problem-solving skills. It鈥檚 not just about finding the value of \(x\); it's also about learning a logical process to handle mathematical challenges efficiently.

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