Problem 68 - Intermediate Algebra

Question

Solve each equation. If the equation is an identity or a contradiction, so indicate. See Example 9. $$ 4(2-3 t)+6 t=-6 t+8 $$

Step-by-Step Solution

Answer

The equation is an identity; it is true for any value of \(t\).

Step 1: Distribute and Simplify

Start by expanding the expression on the left side of the equation. Distribute the 4 across the terms inside the parentheses: \(4(2-3t) = 8 - 12t\). This transforms the equation to \(8 - 12t + 6t = -6t + 8\). Then, combine like terms on the left: \(8 - 6t = -6t + 8\).

Step 2: Compare Both Sides of the Equation

Notice that after simplification, the equation is \(8 - 6t = -6t + 8\). Both sides of the equation are identical, which means the equation holds true for any value of \(t\).

Step 3: Identify the Type of Equation

Since \(8 - 6t = -6t + 8\) is true for any value of \(t\), this equation is an identity.

Key Concepts

Identity Equations

An identity equation is a special type of equation that remains true no matter what values you substitute into the variable. This means that the equation doesn’t rely on the value of the variable to hold true.

In the given exercise, after simplifying the original equation, both sides turned out to be equal (\[8 - 6t = -6t + 8\]), which is a clear indicator of an identity equation.

Here’s why identity equations are important:

• They help confirm that the expression is equivalent in every possible scenario.
• In solving these equations, you often end up with a true statement like \[0 = 0\], which confirms the identity.

Remember, if both sides of the equation are always identical regardless of the variable's value, you're dealing with an identity equation.

Algebraic Simplification

Algebraic simplification involves breaking down complex equations into simpler forms, making it easier to solve them.

Here’s how it was applied in the exercise:

• Start by distributing any constants through expressions in parentheses. The original problem required distributing 4 in the expression \[4(2 - 3t)\] which simplified to \[8 - 12t\].
• Combine like terms to consolidate the equation – in our case, combining \[-12t\] and \[6t\] to \[-6t\].

Simplifying equations is like tidying up a room. It makes everything look clearer and helps in identifying the type of equation you're dealing with faster.

It's a crucial step before solving because it reduces the equation to its essential parts, making it easier to interpret.

Equation Solving Steps

Solving linear equations involves a series of straightforward steps that need careful execution to reach the correct solution.

From the original exercise, these steps were:

• Step 1: Expand and Simplify. Begin by distributing any number outside parentheses to simplify the equation. This may involve combining any like terms as found in the exercise.
• Step 2: Make the Comparison. Once simplified, check if both sides of the equation match, as was done when both sides became identical \[8 - 6t = -6t + 8\].
• Step 3: Identify the Equation Type. Decide whether it's an identity or contradiction once simplified. In this exercise, the equation was always true, classifying it as an identity.

Following these steps ensures that you reach the correct conclusion about what type of equation you're facing. Practicing these steps helps develop a structured approach to solving not only linear equations but more complex algebraic problems too. Always work methodically, check your work, and remember the goal is a balanced equation that is either solved or classified correctly.