Problem 40 - Algebra 1: Concepts and Skills

Question

Find and correct the error. \(2(x-3)=5\) \(2 x-3=5\) \(2 x=8\) \(x=4\)

Step-by-Step Solution

Answer

The error is in the misapplication of the distributive property in the second step. The correct answer is \(x = 5.5\).

Step 1: Identify the Correct Application of the Distributive Property

For \((x-3)\), the multiplication factor '2' should be applied to both 'x' and '-3'. So, \(2(x - 3)\) should be correctly expanded as \(2x - 6\), not \(2x - 3\).

Step 2: Correcting the Wrong Step

Substitute \(2x - 6\) in place of \(2x - 3\) in the second line. Now the equation becomes, \(2x - 6 = 5\).

Step 3: Solve Revised Equation

You then can solve this equation for 'x'. First, isolate 'x' by transferring '-6' to the other side of the equation, which yields \(2x = 5 + 6\), then \(2x = 11\). Now, to get 'x', you divide both sides of the equation by '2', resulting in \(x = 11 / 2\), or \(x = 5.5\).

Key Concepts

Linear Equations

Linear equations are equations of the first degree, meaning the variable appears to the power of one. In their simplest form, they can be represented as \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable we're solving for. Understanding linear equations involves recognizing this basic structure and knowing how to manipulate the equation to find the value of the variable.

In context, the equation \( 2(x - 3) = 5 \) is a linear equation to solve for \( x \). Its goal is to find what value of \( x \) makes the equation true.

Linear equations often involve the distributive property, which allows us to simplify expressions and solve equations efficiently. The distributive property is crucial for expanding brackets and simplifying parts of the equation, as seen in this particular problem.

Solving Equations

The process of solving equations involves a series of steps to find the unknown variable. It's essential to follow a logical sequence and properly apply mathematical operations to both sides of the equation. The key steps for solving linear equations like the one given are:

• Distribute or simplify the equation using properties like the distributive property.
• Isolate the variable by moving constants to the other side of the equation.
• Use operations such as addition or subtraction to eliminate terms.
• Finally, divide or multiply to solve for the variable.

In the corrected equation \( 2(x - 3) = 5 \), we apply these steps:

First, the distributive property is used to expand the equation to \( 2x - 6 = 5 \).

Then, add 6 to both sides to obtain \( 2x = 11 \).

Finally, divide by 2 to solve for \( x \), resulting in \( x = 5.5 \). This linear equation was solved by isolating the variable and performing arithmetic operations.

Error Correction

Identifying and correcting errors is a fundamental skill in mathematics. In the given exercise, a mistake was made during the application of the distributive property. Incorrectly simplifying \( 2(x - 3) \) as \( 2x - 3 \) was the error that needed addressing.

To correct this, we need to ensure the distributive property multiplies the entire expression inside the parenthesis - both the variable \( x \) and the constant -3. The corrected version, \( 2x - 6 \), was obtained by multiplying 2 by both \( x \) and \(-3\).

Spotting such errors often involves:

• Double-checking every step for accuracy.
• Understanding and applying mathematical properties correctly.
• Reviewing calculations to ensure each operation respects the rules of arithmetic.

Developing these skills helps students build confidence in problem-solving and can aid in preventing similar errors in future math exercises.