Problem 47 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Solve the equation symbolically. (b) Classify the equation as a contradiction, an identity, or a conditional equation. $$ \frac{1-2 x}{4}=\frac{3 x-1.5}{-6} $$

Step-by-Step Solution

Answer

Identity; equation holds for all values of \(x\).

Step 1: Eliminate Fractions

In order to eliminate the fractions, we will multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is 12 in this case. After multiplying, the equation becomes: \[ 3(1-2x) = -2(3x-1.5) \] This gives us \[ 3 - 6x = -6x + 3 \] after distributing both sides.

Step 2: Simplify the Equation

Next, simplify the expression by canceling out the like terms on both sides. Add \(6x\) to both sides to eliminate \(-6x\) from the right side of the equation: \[ 3 = 3 \] The variable terms cancel out, and we are left with \(3 = 3\).

Step 3: Classify the Equation

Since the equation simplifies to a true statement \(3 = 3\), this means that the original equation is true for all values of \(x\). Thus, the equation is classified as an identity.

Key Concepts

Identity Equation

An identity equation is one that is true for all possible values of the variable involved. When an equation simplifies to a statement like "3 = 3", as in the original exercise, it means that the expression on both sides of the equation is equivalent for any value of the variable. This classification tells us that there isn't a specific solution for the variable, since every value will satisfy the equation.

• In our original problem, after simplifying, we ended up with no variable but a true statement.
• An identity equation should not be confused with conditional or contradictory equations, which have more specific solutions or none at all.
• The form of an identity equation after simplification typically looks like "a = a" where 'a' is some constant value.

These types of equations are helpful because they confirm that the mathematical expressions involved are perfectly balanced across all conditions.

Eliminating Fractions

Eliminating fractions in an equation can simplify the problem and make it easier to solve. In many cases, working with whole numbers rather than fractions helps clarify the steps required to solve an equation. The common method to clear fractions is to multiply every term of the equation by the least common multiple (LCM) of all denominator values in the equation.

• Start by identifying the denominators in the equation.
• Compute their LCM; for instance, if the denominators were 4 and -6, the LCM is 12.
• Multiply each term of the equation by 12 to eliminate the fractions, changing the equation to have whole numbers only.

By converting the equation into one with whole numbers, we reduce the complexity and make subsequent steps more straightforward. This is a beneficial initial step for many equations, aligning them with simpler arithmetic operations.

Conditional Equation

A conditional equation is true only for specific values of the variable. Unlike an identity equation, a conditional equation features a limited set of solutions that satisfy it. If upon simplifying an equation you isolate the variable and find a specific number (or set of numbers), then the equation is considered conditional.

• Conditional equations often manifest after manipulation or rearrangement leads to finding the exact value of the variable.
• If you arrive at a specific answer such as "x = 5", then the equation is conditional.
• Conditional equations are contrasted with identity equations, which are true for all variable values.

Most equations you'll solve will fall under this category, as they usually describe scenarios or problems with a predefined scope or solution set.