Problem 54 - College Algebra

Question

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

Step-by-Step Solution

Answer

The given equation is an identity.

Step 1: Simplify the Equation

The equation is \(\frac{7 x}{x}=7\). First, simplify the left side of the equation by canceling the \(x\) from the numerator and denominator. This simplifies the equation to \(7=7\).

Step 2: Consider the Equation Type

The equation \(7=7\) is an identity because it is true for all values of \(x\). So, the original equation \(\frac{7 x}{x}=7\) is an identity.

Key Concepts

Conditional Equations

In algebra, it's essential to understand that not all equations are created equal. Conditional equations are a specific type of equation that holds true only under certain conditions or for specific values of the variable. These equations are the bulk of what students typically solve in their math classes, as they work to find the particular value that makes the equation true.

For example, the equation \(x + 5 = 10\) is conditional. It's only true when \(x = 5\). This scenario depends on the value substituting \(x\). Unlike an identity, which is true for any value of \(x\), a conditional equation has one specific solution that satisfies it. When working with these equations, solving often involves:

• Isolating the variable.
• Performing inverse operations to find the solution.
• Verifying the solution by substituting it back into the original equation.

This process strengthens understanding of how changes in values affect equality, helping us build logic and problem-solving skills.

Inconsistent Equations

Inconsistent equations are equations that are never true, no matter what values are substituted for the variables. Simply put, there's no solution that will satisfy the equation. Recognizing an inconsistent equation can save you time and effort during problem-solving by allowing you to quickly identify that an equation has no solution.

A common example of an inconsistent equation is something like \(x + 1 = x\). When you attempt to solve it by subtracting \(x\) from both sides, you get \(1 = 0\). This is a false statement, meaning no value of \(x\) will make the equation true.

In practice, identifying such equations involves:

• Attempting simplification to reach a contradiction.
• Understanding when an equation represents parallel lines in a graph, which never intersect.
• Recognizing patterns of inconsistent results such as expressions equating to different constants.

Knowing about inconsistent equations allows you to effectively dismiss efforts to solve where there's fundamentally no solution possible.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a crucial skill in algebra that makes equations easier to work with by reducing them to their simplest form. This process involves manipulating the expressions according to algebraic rules to eliminate unnecessary complexity and make solving manageable.

The primary steps in simplifying algebraic expressions include:

• Cancelling common factors if present, as seen in the example equation \(\frac{7x}{x} = 7\). By cancelling \(x\), the equation simplifies directly to \(7 = 7\).
• Combining like terms to condense the expression.
• Using distributive property to remove parentheses if needed.

When simplifying, it’s important to ensure you aren’t changing the original value or solution of the expression. Always verify your simplified result by substituting values you think might work back into the equation.

Not only does simplification make finding solutions easier, but it also helps in understanding the structure and relationships within an equation more clearly.