Problem 61 - College Algebra Essentials

Question

Solve equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. \(5 x+9=9(x+1)-4 x\)

Step-by-Step Solution

Answer

The given equation is an identity.

Step 1: Simplify Both Sides

Apply the distributive property on the right side of the equation to remove parenthesis: \$9(x+1) = 9x + 9\$ and simplify the equation to: \(5x + 9 = 9x + 9 - 4x\). This then simplifies to: \(5x + 9 = 5x + 9\).

Step 2: Subtract 5x from Both Sides

Subtract 5x from both sides to isolate the variable x on one side which results into: \(9 = 9\).

Step 3: Classify the Equation

The equation \(9=9\) is always true, irrespective of the value of x, so the given equation is an identity.

Key Concepts

Identity Equation

An identity equation is a type of equation that holds true for all possible values of the variable. This means no matter what number you substitute for the variable, the equation will always be balanced. In our example, we arrived at the expression \(9 = 9\), which is true for any value of \(x\).

Identity equations are significant because they show a relationship that is universally valid.

• For instance, the equation \( x + 5 = 5 + x \) is an identity because it simplifies to \(0 = 0\) after subtracting \(x + 5\) from both sides.
• These equations often appear when simplifying expressions, and they signal that the expressions on both sides of the equation are equivalent.

Remember: An identity equation doesn't provide specific solutions, but instead reinforces the concept that both sides are identical by nature.

Conditional Equation

A conditional equation, on the other hand, is true only for specific values of the variable. This type of equation is what we typically solve in algebra because it gives us a particular solution or solutions.

For example, if you have the equation \(2x + 3 = 11\), you solve for \(x\) to find that \(x = 4\). This is a conditional equation because it only holds for \(x = 4\).

• They are called 'conditional' because the truth of the equation depends on the condition or value of the variable.
• When solving, once isolated, they typically result in an expression such as \(x = a\), where \(a\) is the solution.

These offer practical use as they help us find specific values that satisfy certain conditions or situations.

Inconsistent Equation

Inconsistent equations are equations that do not have a solution. No matter what value you input for the variable, the equation remains unbalanced. This occurs because the expressions on both sides of the equation are fundamentally contradictory.

For example, consider \(2x + 5 = 2x - 3\). When we attempt to solve it, by subtracting \(2x\) from both sides, we end up with \(5 = -3\), a false statement, indicating an inconsistent equation.

• Inconsistent equations are useful in understanding limitations and discrepancies in problem settings.
• They teach us that not every algebraic equation has a solution, highlighting constraints within mathematical systems.

Recognizing inconsistency is important because it signals that the relationship described by the equation cannot exist within the given parameters.