Problem 61 - Algebra and Trigonometry

Question

Solve each 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 equation \(5x + 9 = 9(x + 1) - 4x\) simplifies to \(5x + 9 = 5x + 9\), which holds true for any value of x. Thus, this equation is an identity.

Step 1: Simplifying the Equation

The first step is to simplify the equation by distributing and combining like terms. Distribute the 9 to the terms inside the bracket on the right-hand side of the equation to get: \(9x + 9\). Now the equation is \(5x + 9 = 9x + 9 - 4x\). Combine the like terms on the right-hand side to simplify further: \(5x + 9 = 5x + 9\).

Step 2: Checking the Possible Type of the Equation

Now the equation is \(5x + 9 = 5x + 9\). We see that this is true for any and all values of x. This means the given equation is an identity because an identity is an equation that is always true for any value of the variable.

Key Concepts

Understanding Identity Equations

An identity equation is one of the fundamental types of equations in algebra. This type of equation is equal on both sides for any value of the variable involved. For instance, take the equation from our exercise: \(5x + 9 = 5x + 9\). As you can see, if you substitute any value for \(x\), the equation remains balanced. Here are a few characteristics of identity equations:

• The equation holds true for all permissible values of the variable.
• After simplifying, both sides of the equation look exactly the same.
• Examples include equations like \(x + 3 = x + 3\) or \(2y - y = y\).

Understanding identity equations is crucial, as it helps recognize when an equation is always valid irrespective of variable values. These equations often serve as verification tools in algebra, demonstrating inherent truths.

Exploring Conditional Equations

Conditional equations offer another intriguing aspect of solving equations. These equations are true only for certain values of the variable, unlike identity equations. Let's break down how conditional equations work:

• After simplifying, one side of the equation does not equal the other for all values.
• Conditional equations have specific solutions, often a single value or set of values.
• For example, \(2x + 3 = 7\) is a conditional equation, which only holds true when \(x = 2\).

When working with such equations, solving it involves isolating the variable to identify which specific value makes the equation true. It's a bit like solving a puzzle, where you find the exact key that fits the lock. It's essential to check these solutions to confirm their validity.

In-depth Look at Inconsistent Equations

Inconsistent equations represent scenarios where no solution exists. This means there are no values of the variable that will satisfy the equation making both sides equal.Here's what to know about inconsistent equations:

• After simplifying, they result in a contradiction such as \(0 = 5\) or \(1 = 2\).
• These equations highlight impossibilities in mathematical problems.
• For example, the equation \(x + 2 = x + 3\) is inconsistent because both sides can never be equal for any value of \(x\).

When you come across such equations, it is important to recognize that it is not a sign of a mistake in arithmetic, but rather a natural result indicating that no variable value can meet the equation's conditions.