Problem 40 - Algebra 1: Concepts and Skills

Question

find and correct the error. $$\begin{aligned}3(x-2)+5 x &=3(6 x-2) \\\&=18 x-6\end{aligned}$$

Step-by-Step Solution

Answer

The correct form of the equation is \(3(x - 2) + 5x = 18x - 6\).

Step 1: Identify the error in the equation

Looking at the equation, it appears that the mistake is in the distribution of the 3 inside the bracket on the right side of the equation. The calculated result, \(18x - 6\), is incorrect. Using the distributive property, it should be \(3 \times 6x - 3 \times 2\).

Step 2: Correct the error

Applying the distributive property correctly, \(3 \times 6x = 18x\) and \(3 \times 2 = 6\). So we have \(18x - 6\) which corrects the error.

Step 3: Rewrite the entire equation

Substituting the correct values into the equation, it becomes \(3(x - 2) + 5x = 18x - 6\). This is now the corrected version of the equation.

Key Concepts

Understanding Algebraic Equations

Algebraic equations are the backbone of algebra and are essential for solving various mathematical problems. An equation represents a balance between two expressions; if you perform the same operation on both sides, the balance remains unchanged. However, understanding and solving equations require a strong grasp of algebraic laws, including the fundamental operations and properties like the distributive property.

Consider an algebraic equation as a scale that must be kept in equilibrium. Just as adding or removing weights from one side of the scale requires an identical action on the other to maintain balance, each step in solving an equation requires that both sides be treated equally. When approaching algebraic equations, keep the objective in mind: isolate the variable to find its value, which effectively 'solves' the equation.

Error Correction in Algebra

Error correction in algebra involves scrutinizing each step of a calculation to uncover and correct mistakes. Common errors include miscalculating operations, misapplying algebraic properties, and oversights in simplification. Mistakes can undermine the accuracy of the solution, making error correction a critical skill for any student.

Here's a useful checklist for error correction:

• Verify that all operations comply with algebraic rules.
• Reassess each step to catch any missed errors.
• Doubles check the final answer by plugging it back into the original equation.

Through diligent practice and attention to detail, students can minimize errors and ensure they apply algebraic principles correctly.

Applying the Distributive Property

The distributive property is a powerful tool in algebra that allows us to simplify expressions by distributing a factor across terms within parentheses. Mathematically, it is expressed as \(a(b + c) = ab + ac\). This rule is essential when expanding algebraic expressions or solving equations.

Incorrect application of the distributive property is a common error. To avoid this, always multiply the term outside the parentheses by each term inside individually. For instance, if you come across an expression like \(3(x - 2) + 5x\), you would apply the distributive property to the first part to get \(3x - 6 + 5x\), then combine like terms. Clear understanding and proper utilization of the distributive property aids greatly in algebraic problem-solving.