Problem 31 - Prealgebra

Question

Write two different equations whose solutions are each 5 .

Step-by-Step Solution

Answer

Equations: 1. \( x = 5 \), 2. \( 2x - 5 = 5 \).

Step 1: Understand the Requirement

The exercise asks us to create two different equations that have the solution 5. This means we need to set up two mathematical expressions that, when solved, both give the result of 5.

Step 2: Create the First Equation

Let's consider the simple equation of the form \( x = a \), where \( a = 5 \). Thus, the first equation is \( x = 5 \). If \( x = 5 \), then the solution is directly 5.

Step 3: Create the Second Equation

For the second equation, we can use a linear equation. Consider \( 2x - 5 = 5 \). Solve this equation: Add 5 to both sides to get \( 2x = 10 \), then divide both sides by 2 to get \( x = 5 \). Thus, this equation also has the solution 5.

Key Concepts

Understanding Linear Equations

Linear equations play a crucial role in mathematics as they represent relationships between variables that have a constant rate of change. They are

• Equations with variables raised to the power of one.
• Straight line equations when graphed on a coordinate plane.

In the context of our exercise, we used two linear equations:

• The very simple case: \( x = 5 \), which is straightforward with no additional operations. This showcases the simplest linear relationship: the variable equals a constant.
• The more involved equation: \( 2x - 5 = 5 \). Here, the term \( 2x \) represents the variable being doubled before subtracting 5.

In general, solving linear equations involves simple arithmetic operations, making them a good starting point for solving more complex mathematical problems.

Finding the Solution of Equations

The process of finding solutions for equations involves determining the value of the variable that makes the equation true. In other words, it's about identifying the number that satisfies the mathematical sentence. The steps taken to solve the second equation \( 2x - 5 = 5 \) included a few fundamental principles:

• Isolate the variable by performing inverse operations. For \( 2x - 5 = 5 \), first, add 5 to both sides of the equation.
• From \( 2x = 10 \), divide each side by 2 to isolate \( x \), yielding \( x = 5 \). This step-by-step simplification pinpoints the sought value.

The solution is the number that, when substituted back into the original equation, maintains equality. Confirming each step through substitution not only helps check correctness but reinforces understanding of operations applied.

The Role of Mathematical Expressions

Mathematical expressions are the building blocks of equations and calculations. They consist of constants, variables, and mathematical operations - such as addition or multiplication. In our solved exercise:

• \( x = 5 \) is an expression where the variable equals a constant without additional calculations needed.
• The expression \( 2x - 5 = 5 \) is more involved due to extra arithmetic operations.

Every piece of an expression, like \( 2x \) or \( -5 \), contributes to the overall mathematical relationship. Understanding each component's role within an expression aids in grasping the full equation and the problem-solving process itself. Mathematical expressions act as the language through which mathematics explains and solves reality's problems.