Problem 76 - College Algebra

Question

combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. $$ \frac{3}{x}-\frac{1}{6}=\frac{1}{3} $$

Step-by-Step Solution

Answer

The solution is x = 6 and it is a conditional equation.

Step 1: Clear the fractions

It's usually easier to work with equations without fractions. To remove the fractions, find the least common denominator (LCD) of x and 6, which is 6x. Now, multiply both sides of the equation by 6x.

Step 2: Simplify the Equation

After multiplication, the equation becomes: 18 - x = 2x. This simplifies to 3x = 18.

Step 3: Solve for x

Divide both sides by 3 to get x = 6.

Step 4: Verify the solution

Substitute x = 6 in the original equation.

Step 5: State the nature of the equation

After verification, if x = 6 gives true equality, the equation is conditional equation. If every value of x tends to give true equality, the equation is an identity. If there is no possible value that gives true equality, the equation is inconsistent.

Key Concepts

Identity

In algebra, an **identity** is an equation that holds true for all possible values of the variable involved. This means that no matter what number you substitute for the variable, the equation will always balance out as true. You can think of it like a universal truth in mathematics. For example, the equation \( x - x = 0 \) is an identity because regardless of what \( x \) you use, the equation will always equal zero. Identities serve as useful tools in algebra for simplifying expressions and solving various mathematical problems.
When identifying an equation as an identity, always remember to check whether substituting any value for the variable results in a true statement. For identities, this will always be the case. Hence, identities don't depend on a specific solution; they represent a consistent truth across all numbers.

Conditional Equation

A **conditional equation** is true only for specific values of the variables involved. Unlike an identity, a conditional equation does not hold for all possible numbers. Instead, there is one or a few numbers that satisfy the equation.
For example, consider the equation \( 2x + 3 = 7 \). To find out the specific value(s) that make this equation true, we solve it for \( x \). In this case, \( x = 2 \) will satisfy the equation. When \( x \) equals any other number, the equation does not hold.

• Conditional equations often appear when solving algebraic problems because they reveal particular values that satisfy given conditions.
• These equations are very common in algebra, and solving them requires manipulating the equation to isolate the variable.
• Once you solve for the variable, verifying your solution by substituting it back into the original equation ensures accuracy.

Conditional equations play a crucial role in algebra, guiding you through processes of finding solutions that fit certain criteria.

Inconsistent Equation

An **inconsistent equation** is one that has no solution. In other words, there is no possible value for the variable that makes the equation true. These equations lead to contradictions if you attempt to solve them.
For instance, consider \( x + 3 = x - 2 \). By trying to solve this equation by eliminating \( x \), you end up with the statement \( 3 = -2 \), which is clearly impossible. Therefore, the equation is inconsistent. This type of equation can emerge when attempting to solve systems of equations that don't intersect or share any common solutions.

• Recognizing an inconsistent equation involves simplifying and analyzing whether the results lead to an impossible statement.
• Inconsistent equations are important to identify because they indicate an inherent conflict within the conditions of the problem.
• Understanding when an equation is inconsistent is crucial for correctly interpreting mathematical results and problem conditions.

When you encounter such equations, it's essential to acknowledge the lack of solutions and consider whether the equations or conditions might need to be modified or re-considered to achieve a correct solution.