Problem 15 - Algebra for College Students

Question

This incorrect solution contains a common error. \(\begin{aligned} 8 x-2(2 x-3) &=3 x+7 \\ 8 x-4 x-6 &=3 x+7 \text { Distributive property }\\\ &4 x-6=3 x+7 \quad \text {Combine like terms}\\\ &x=13 \quad \text {Subtract 3 x . Add 6} . \end{aligned}\) WHAT WENT WRONG? Give the correct solution.

Step-by-Step Solution

Answer

\(x = 1\)

Step 1: Distribute - Correctly apply distributive property

Use the distributive property to correctly simplify the left side of the equation: \text 2(2x-3) \text Would become 4x+6 So the correct expression should be: \[8x - 2(2x - 3) = 3x + 7\]becomes \[8x - 4x + 6 = 3x + 7\]

Step 2: Simplify - Combine like terms

Combine the like terms on the left side: \[ (8x - 4x) \] becomes \[4x + 6 = 3x + 7\].

Step 3: Move variable terms

Subtract \(3x\) from both sides to get all the \(x\) terms on one side: \[(4x - 3x) + 6 = 7\], which simplifies to \[x + 6 = 7\].

Step 4: Isolate the variable

Subtract \(6\) from both sides to solve for \(x\): \[x + 6 - 6 = 7 - 6\], which simplifies to: \[x = 1\].

Key Concepts

Distributive Property

The distributive property is an essential tool in solving linear equations. It allows us to simplify expressions by distributing a factor across terms inside parentheses. For example, in the expression \(2(2x - 3)\), we distribute the 2 to both terms inside the parentheses:

\[2 \times 2x + 2 \times (-3) = 4x - 6\]

This correctly applies the distributive property. The error in the original solution was not distributing correctly. Remember always to multiply the factor by each term within the parentheses.

Combine Like Terms

Combining like terms is a crucial step after distributing in simplifying expressions. Like terms have the same variables raised to the same power. In our problem, after correctly distributing, we simplify the equation:

\[8x - 4x - 6 = 3x + 7\]

We combine the \(8x\) and \(-4x\) on the left side because they are like terms. This results in:

\[4x - 6 = 3x + 7\]

Combining terms correctly keeps the equation balanced and simpler.

Isolating Variables

Isolating the variable is the process of getting the variable of interest (usually \(x\)) by itself on one side of the equation. We do this by performing operations that allow us to move other terms to the opposite side. After combining like terms, we had:

\[4x - 6 = 3x + 7\]

We can subtract \(3x\) from both sides to get all x terms on one side:

\[4x - 3x - 6 = 7\]

This simplifies to:

\[x - 6 = 7\]

We then add 6 to both sides to isolate \(x\):

\[x = 13\]

Correctly isolating the variable is crucial for finding the solution.

Linear Equations

Linear equations involve variables to the first power and their solutions form a straight line when graphed. A linear equation typically has the form \(ax + b = c\). In our problem, we are dealing with such an equation:

\[8x - 2(2x - 3) = 3x + 7\]

Simplifying and solving this equation involves applying the distributive property, combining like terms, and isolating the variable. The correct solution reveals that linear equations are solved through a series of logical steps, all designed to find the value of \(x\). Remember, understanding these steps improves your ability to handle any linear equation accurately.