Problem 67 - Intermediate Algebra

Question

Solve each equation. If the equation is an identity or a contradiction, so indicate. See Example 9. $$ 8 x+3(2-x)=5 x+6 $$

Step-by-Step Solution

Answer

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

Step 1: Distribute and Simplify

First, distribute the 3 over the terms inside the parentheses: \(3(2-x)\). This gives \(3 \times 2 = 6\) and \(3 \times (-x) = -3x\). So the equation becomes: \[8x + 6 - 3x = 5x + 6\]. Combine like terms on the left side by subtracting \(3x\) from \(8x\): \[5x + 6 = 5x + 6\].

Step 2: Isolate Variable Terms

To isolate the variable terms, subtract \(5x\) from both sides of the equation. This results in: \[5x + 6 - 5x = 5x + 6 - 5x\], which simplifies to: \[6 = 6\].

Step 3: Analyze Resulting Equality

Since the resulting equality \(6 = 6\) is a true statement regardless of the value of \(x\), the original equation is an identity. This means that the original equation is true for all values of \(x\). No specific solution for \(x\) is required.

Key Concepts

Identities in Algebra

Identities in algebra are equations that hold true for any value of the variable. In the context of solving linear equations, recognizing an identity is significant because it tells us that no specific values solve the equation—they all do! For instance, when solving an equation and reducing it to a statement like \(6 = 6\), which is undoubtedly true, you've encountered an identity.

Here’s why this happens: starting with the original equation, you simplify and manipulate it using algebraic steps such as combining terms or isolating variables. If the variables cancel out and you end up with a universally true statement, the equation is not conditional upon any particular value of the variable.

• Identities indicate the solution set includes all real numbers.
• Contrary to an identity, a contradiction results in a false statement like \(6 = 5\), which implies no solution.

Understanding identities in algebra helps you recognize when an equation holds without finding specific solutions, streamlining the solving process.

Distributive Property

The distributive property is a key tool in algebra that allows you to simplify expressions and solve equations effectively. This property states that multiplying a single term by a sum or difference is the same as multiplying the term by each component separately and then adding or subtracting the results.

A common example is the expression \(a(b + c)\), which can be expanded as \(ab + ac\). This principle was applied in the original exercise by distributing the \(3\) to both \(2\) and \(-x\) within the parentheses. It transformed \(3(2-x)\) into \(6 - 3x\).

• Effective for simplifying equations by removing parentheses.
• Can be applied to both addition and subtraction within parentheses.

Leveraging the distributive property simplifies the process of combining like terms—a crucial next step in solving linear equations.

Combining Like Terms

Combining like terms is a crucial step in simplifying algebraic expressions and solving equations. It involves merging terms that have the same variable raised to the same power. By doing so, you reduce the number of terms and simplify the expression.

In the equation \(8x + 6 - 3x = 5x + 6\), like terms include \(8x\) and \(-3x\), which combine to form \(5x\). This essential maneuver not only simplifies the expression but also helps isolate terms involving variables on one side of the equation.

• Makes equations simpler and easier to solve.
• Always pay attention to the signs when combining terms.

Practicing the art of combining like terms will enhance your problem-solving efficiency in algebra.