Problem 74 - Elementary Algebra

Question

Translate the following sentences into linear equations and then solve. The sum of \(-3 x\) and 7 is equal to 14 .

Step-by-Step Solution

Answer

The value of \(x\) is \(-\frac{7}{3}\).

Step 1: Identify the Expression

The exercise states that the sum of \(-3x\) and 7 is equal to 14. This can be expressed in an equation format.

Step 2: Write the Equation

From the expression "the sum of \(-3x\) and 7 is equal to 14", we can write the equation as: \(-3x + 7 = 14\).

Step 3: Isolate the Variable Term

Subtract 7 from both sides of the equation to isolate the term with the variable. This gives: \(-3x + 7 - 7 = 14 - 7\), which simplifies to \(-3x = 7\).

Step 4: Solve for the Variable

Divide both sides by \(-3\) to solve for \(x\). This is done as follows: \(x = \frac{7}{-3}\), which simplifies to \(x = -\frac{7}{3}\).

Key Concepts

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations like addition or subtraction. They bring together numbers and letters (variables) to represent real-world problems mathematically.
For instance, in this exercise, the expression is "the sum of \(-3x\) and 7". Here, \(x\) is the variable, which represents an unknown number that we are trying to find.

• The term \(-3x\) suggests that \(x\) is multiplied by -3.
• Adding 7 to it reflects another operation on this expression.

By translating a spoken problem into an algebraic expression, we can use mathematical operations to solve for the unknown, which can be crucial for tackling more complex problems later on.

Solving Equations

Solving equations involves finding the value of the unknown variable that makes the equation true. It's a combination of understanding the operation in the equation and performing it in a way that maintains equality.
In the given exercise, we start with the equation \(-3x + 7 = 14\). Our goal here is to make sure both sides of the equation reflect the same value.

• This begins with applying inverse operations, like subtraction or division, to both sides of the equation.
• The aim is to simplify the equation step by step until the variable stands alone on one side.

Solving equations is a key skill in mathematics because it allows us to find specific solutions to various problems.

Isolating Variables

Isolating the variable is a crucial step in solving linear equations. It means manipulating the equation such that the variable ends up alone on one side of the equation.
In our exercise, starting with \(-3x + 7 = 14\), we need to isolate \(x\) by removing every other term surrounding it.

• First, subtract 7 from both sides to eliminate the constant term on the left.
• This simplifies our equation to \(-3x = 7\).

Isolating the variable helps simplify complex problems and makes it easier to solve, especially because it lays the foundation for more advanced algebraic techniques.

Equation Format

Equations can appear in various formats, but in algebra, the standard form involves terms that are clearly defined on either side of the equality sign. This can be seen in the format \(ax + b = c\), where \(a\), \(b\), and \(c\) are numbers and \(x\) is the variable.
For this exercise, the equation \(-3x + 7 = 14\) fits into this standardized linear equation format:

• \(-3x\) is the term containing the variable.
• 7 is the constant added to it, and 14 is the number it equals.

Recognizing and understanding the format of equations simplifies the process of solving them. It provides a structured way to approach equations and ensures we apply operations correctly to both sides.