Problem 73 - Algebra A Combined Function

Question

a. Solve: \(\quad x+3=x+3\) b. If you simplify an equation (such as the one in part a) and get a true statement such as \(3=3\) or \(0=0\), what can you conclude about the solution(s) of the original equation? C. On your own, construct an equation for which every real number is a solution.

Step-by-Step Solution

Answer

a. Any real number; b. Every real number is a solution; c. Example: \(5(x + 2) = 5x + 10\).

Step 1: Analyzing the Equation

First, let's take a look at the equation: \(x+3 = x+3\). This equation has the same terms on both sides, which means they are identical.

Step 2: Identify Simplification

Simplify the equation by subtracting \(x\) from both sides to see what remains:\[x + 3 - x = x + 3 - x\]Resulting in:\[3 = 3\]

Step 3: Understanding the Result

The equation \(3 = 3\) is always true. This indicates that no matter what value you substitute for \(x\), the equation remains balanced. Thus, every real number can be a solution since both sides are equal regardless of \(x\).

Step 4: Conclusion from Simplification

When simplifying leads to a true statement like \(3 = 3\), it means that the original equation holds true for all values of \(x\). Thus, the solution to \(x + 3 = x + 3\) is any real number. You conclude that every real number is a solution.

Step 5: Constructing an Equation with Unrestricted Solutions

Create a similar equation to part a, where the terms on both sides will always satisfy the equality regardless of the variable's value. For example:\[5(x + 2) = 5x + 10\]Expanding the left side gives \(5x + 10\), which matches the right side exactly, thus true for all \(x\).

Key Concepts

Identical Equations

When you encounter equations like \( x+3 = x+3 \), these are known as identical equations. Identical equations are those in which the expressions on both sides of the equality sign are exactly the same. This means that whatever value is substituted for the variable, the equation will always remain true.
For example, when you simplify \( x+3 - x = x+3 - x \), you end up with \( 3 = 3 \), a clear indication that the equation is identical.
Identical equations are simple but fundamental to understanding more complex algebraic concepts. They teach us that:

• If subtracting the same quantity from both sides results in an equivalent expression (like \( 3=3 \)), then the equation has infinite solutions.
• Such an equation is always true, representing an identity rather than a conditional equation.

True Statements

True statements occur when the simplification of an equation leads to an equality that holds regardless of the variable involved. A typical example is \( 3 = 3 \). This statement is always true, no matter what number you replace the variable with or even if no variable is present.
When dealing with algebra, recognizing true statements can simplify the problem-solving process. Here's what you need to remember about them:

• They signal that the equation is valid for all possible variable values, not just for a specific solution.
• These are "tautologies" in logical terms, meaning they do not hinge on any specific condition or variable.
• Understanding that a simplified statement like \( 0 = 0 \) or \( 5 = 5 \) always holds allows students to identify when an algebraic equation is a broad identity.

By practicing identifying these true statements, you become more adept at solving algebraic equations efficiently.

Solutions of Equations

The solutions of equations are the values of variables that make an equation true. Identical equations like \( x+3=x+3 \) illustrate a scenario where every real number could be a solution.
When an equation simplifies to an always-true statement like \( 3=3 \), it means the equation holds true no matter what the variable is, implying infinite solutions.
Here's a breakdown of how solutions can vary:

• **Single Solution:** An equation like \( x + 2 = 5 \) has a single solution \( x = 3 \), resulting from balanced operations.
• **No Solution:** Contradictory equations, like \( x + 3 = x + 5 \) simplify to an impossible statement \( 3 = 5 \), indicating no solution exists.
• **Infinite Solutions:** When simplification results in a true statement like \( 3 = 3 \), the original equation is valid for all values of the variable. This case indicates infinite solutions.

Understanding different types of solutions empowers students to analyze and solve equations with greater confidence and skill.