Problem 34 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The equation is an identity, true for all \( t \).

Step 1: Distribute the 4

Start by distributing the 4 in the expression on the left side of the equation: \[ 4(2) - 4(3t) = 8 - 12t \]Thus, the equation becomes: \[ 8 - 12t + 6t = -6t + 8 \]

Step 2: Combine like terms on the left side

Combine the terms with \( t \) on the left side of the equation: \[ 8 - 12t + 6t = 8 - 6t \]This simplifies to: \[ 8 - 6t = 8 - 6t \]

Step 3: Analyze the equation

Notice that both sides of the equation are identical: \[ 8 - 6t = 8 - 6t \] This means the equation is true for all values of \( t \).

Key Concepts

Understanding Identity Equations

An identity equation is a special type of equation that is always true, no matter what values are chosen for the variable(s) involved. Think of them as universal truths within the scope of algebra. For instance, if you simplify an equation and end up with something like \( x + 3 = x + 3 \), notice how both sides are exactly the same. This is a key indicator of an identity. In our example problem, we have \( 8 - 6t = 8 - 6t \). Observing that both sides of the equation are identical automatically tells us that the equation holds true for any value of \( t \). Essentially, there isn't just one solution. Every possible value of \( t \) is a solution.Identity equations are important because they highlight underlying relationships between algebraic expressions. Recognizing these equations can simplify solving and understanding algebraic problems.

Distributing Terms in an Equation

Distributing or distribution in algebra refers to the process of multiplying a single term by each term inside parentheses. This is done using the distributive property, which states that \( a(b + c) = ab + ac \). This property allows you to remove the parentheses and simplify the expression.In our example, we initially have \( 4(2-3t) \). Applying the distributive property looks like this:

• Multiply 4 by 2 to get 8.
• Multiply 4 by \(-3t\) to get \(-12t\).

After distribution, the expression becomes \( 8 - 12t \). When you distribute terms correctly, you help prepare the equation for the next steps, such as combining like terms. This simplification step is crucial as it makes equations easier to solve or analyze.

Combining Like Terms

Combining like terms is another essential algebraic skill. This involves grouping terms that have the same variable part. Usually, these terms are added or subtracted from each other to form a single term.In our equation, after distribution, we reached \( 8 - 12t + 6t \). Notice the terms \(-12t\) and \(6t\) both involve \(t\) and thus are like terms. To combine them:

• Add (or subtract) their coefficients: \(-12 + 6 = -6\).
• Combine them to form \(-6t\).

After combining, the simplified equation is \( 8 - 6t \). This simplification step is vital as it reduces the equation’s complexity, helping us more easily see patterns or solutions, such as discovering that it is an identity equation.