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Chapter 5: Problem 69

Solve the following equations. See Section 2.1 $$ 3 x+1=0 $$

Short Answer

Expert verified
\(x = -\frac{1}{3}\)

Step by step solution

01

Isolate the Variable

Start by isolating the variable term on one side of the equation. We want to get rid of the constant term on the left side, so subtract 1 from both sides:\[3x + 1 - 1 = 0 - 1\]This simplifies to:\[3x = -1\]
02

Solve for the Variable

Now, solve for \(x\) by dividing both sides of the equation by 3. This will give us:\[x = \frac{-1}{3}\]Thus, the solution for the equation is \(x = -\frac{1}{3}\).

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Key Concepts

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

Solving Equations
Solving equations is a fundamental skill in mathematics, and it involves finding the value of a variable that makes an equation true. This process is essential because it helps in understanding relationships between different quantities.
To solve an equation like \(3x + 1 = 0\), we need to perform operations that will isolate the variable. These operations often include addition, subtraction, multiplication, or division, which are done to both sides of the equation to maintain balance.
  • Identify the variable you need to solve for, which is usually represented by letters like \(x\), \(y\), etc.
  • Perform operations to simplify and solve the equation.
Understanding the process of solving equations enhances problem-solving skills in various areas such as algebra, calculus, and even in real-world scenarios.
Isolating Variables
Isolating variables is a key step when solving equations. It means rearranging the equation so that the variable is alone on one side.
For the equation \(3x + 1 = 0\), we want to isolate \(x\). We begin by removing any constants or coefficients attached to \(x\).
In our example, this means subtracting 1 from both sides, resulting in \(3x = -1\). By doing this, we can focus on \(x\) and make further steps to solve for it.
  • Remove constants by performing arithmetic operations like addition or subtraction.
  • Divide or multiply to eliminate coefficients next to the variable.
Once isolated, you can clearly see what additional steps are needed to find the solution of the equation.
Equation Simplification
Equation simplification involves reducing an equation to its simplest form. The goal is to make the equation easier to understand and solve.
When we simplified \(3x + 1 = 0\), we first subtracted 1 from both sides to eliminate the constant, resulting in \(3x = -1\). This step reduces the equation's complexity.
Now, you divide both sides by 3 to completely isolate \(x\), simplifying the equation to \(x = \frac{-1}{3}\).
  • Simplification removes unnecessary elements to reveal the simplest form of the equation.
  • It often involves performing inverse operations to undo additions, subtractions, multiplications, or divisions.
This process is crucial for ensuring equations are manageable and systematically lead to finding the correct solution.

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Most popular questions from this chapter

Factor each polynomial completely. See Examples 1 through 12. $$ 2 x^{2}+16 x y+32 y^{2} $$

Find the value of \(c\) that makes each trinomial a perfect square trinomial. $$ m^{2}-14 m+c $$

Factor. Assume that variables used as exponents represent positive integers. $$ 9 x^{2 n}+24 x^{n}+16 $$

Factor. Assume that variables used as exponents represent positive integers. $$ 4 x^{2 n}-12 x^{n}+9 $$

Factor each polynomial completely. See Examples 1 through 12. $$ 3 a^{2}+12 a b+12 b^{2} $$
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