Problem 51 - College Algebra

Question

In Exercises \(51-58,\) determine whether each equation is an identity, a conditional equation, or an inconsistent equation. $$ 4(x-7)=4 x-28 $$

Step-by-Step Solution

Answer

The given equation is an identity because it holds true for all real numbers.

Step 1: Distribute the multiplication

Begin by distributing the multiplication on the left hand side to eliminate parentheses: \(4x - 28\). Now, the equation looks like \(4x - 28 = 4x - 28\).

Step 2: Analyze the equation

From Step 1, you can observe that both sides of the equation are identical. This means that no matter what real number value we substitute for \(x\), both sides of the equation will remain equal.

Key Concepts

Conditional Equations

A conditional equation is an equation that holds true only for specific values of the variable involved. It's like a puzzle that only fits when you have the right piece. In mathematical terms, a conditional equation is true only for certain values of its variable, and false for others.

For example, consider the equation \(x + 2 = 5\). This equation is considered conditional because it will only be true when \(x = 3\). For any other value of \(x\), the equation won't balance, making it false.

• If you replace \(x\) with 3, both sides become equal: \(3 + 2 = 5\).
• Unlike identity equations, conditional equations only work for specific numbers.

Remember, when solving a conditional equation, you're finding that one unique piece - or value - that makes the entire puzzle come together smoothly.

Inconsistent Equations

Inconsistent equations pose quite a challenge, as these are equations that are never true. No matter what value you assign to the variable, the equation will not hold. It's like trying to fit a square peg into a round hole - it just won't work.

These equations have no solution because they set up an equality that cannot be satisfied. A simple example is the equation \(x + 2 = x + 3\). No value for \(x\) will make the two sides equal.

• For \(x + 2\), even if \(x\) is 0, 1, or any other number, the right side \(x + 3\) will always be greater by 1.
• Inconsistent equations often highlight contradictions or unresolvable conflicts in algebraic expressions.

Understanding inconsistent equations can help you quickly identify when an algebraic problem has no feasible solution.

Algebraic Equations

Algebraic equations are vital components of algebra, representing statements where two expressions are set equal to one another. They typically include one or more variables, constants, and mathematical operations. Algebraic equations come in different forms:

• Linear equations, such as \(2x + 3 = 7\), have one variable raised to the first power.
• Quadratic equations, like \(x^2 - 4x + 4 = 0\), have variables raised to the second power.
• Polynomial equations may include higher powers, such as \(x^3 + 2x^2 - x - 5 = 0\).

Each type of algebraic equation requires different methods for finding solutions. For example, linear equations can be solved by isolating the variable, while quadratic equations might require factoring or the quadratic formula.

Algebraic equations form the backbone of algebra, offering powerful tools to model and solve real-world problems. They are an essential part of mathematics, helping us explore relationships and make predictions based on given data.