Problem 53 - Algebra 2

Question

Solve: \(5(x+4)=x+4\) \(\begin{array}{ll}{\text { Step } 1 : 5 x+20} & {=x+4} \\ {\text { Step } 2 :} & {4 x+20=4} \\ {\text { Step } 3 :} & {4 x=24} \\ {\text { Step } 4 :} & {x=6}\end{array}\) Which is the first incorrrect step in the solution shown above? A. Step 4 B. Step 3 C. Step 2 D. Step 1

Step-by-Step Solution

Answer

Step 3 is incorrect.

Step 1: Applying the distributive property

Start by applying the distributive property to the left side of the equation: \[5(x+4) = 5 \cdot x + 5 \cdot 4 = 5x + 20\]Therefore, the equation becomes:\[5x + 20 = x + 4\]

Step 2: Simplify the equation

Subtract \(x\) from both sides to begin isolating the variable:\[5x + 20 - x = x + 4 - x\]This simplifies the equation to:\[4x + 20 = 4\]

Step 3: Isolate terms involving \(x\)

Subtract 20 from both sides to continue simplifying and isolate the terms with \(x\):\[4x + 20 - 20 = 4 - 20\]This gives us:\[4x = -16\]

Step 4: Solve for \(x\)

Divide both sides by 4 to solve for \(x\):\[\frac{4x}{4} = \frac{-16}{4}\]Thus, \(x = -4\).

Key Concepts

Equation Solving

Understanding equation solving means finding the variable value that makes a mathematical statement true. In our example, we're trying to find the value of \(x\) that solves the equation \(5(x+4) = x + 4\). Solving equations often involves several steps:

• Distributive property: Multiply a single term across terms within parentheses to eliminate parentheses.
• Simplifying: Combine like terms to create a simpler equation.
• Isolating the variable: Perform operations to get the variable on one side of the equation by itself.
• Solving: Finalize the calculation to find the variable's value.

These systematic steps help ensure the solution is accurate, as seen in equation solving.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations. In our equation, \(5(x+4)\), the expression includes:

• Numbers: These are constants, like 5 and 4, which remain fixed.
• Variables: Represent unknowns, like \(x\), which we aim to solve.
• Operators: Symbols indicating operations, such as multiplication and addition.

In algebra, handling expressions involves understanding how these components interact. The distributive property, for instance, is vital in distributing a number across a sum within parentheses to create a workable equation. Recognizing how to manipulate these expressions is crucial in algebra.

Simplification Process

The simplification process involves making an equation easier to handle without changing its solution. Here, it means reducing the equation so it's easier to isolate the variable. Let's break down the process used:

• Apply the distributive property: To transform \(5(x+4)\) into \(5x + 20\).
• Combine like terms: To focus on the variable, subtract one \(x\) from both sides, simplifying from \(5x + 20 = x + 4\) to \(4x + 20 = 4\).
• Isolate the variable: Further simplify by subtracting constants, leading to \(4x = -16\).

This effective process is crucial to solving any algebraic equation, as it breaks down complex expressions into more manageable parts, leading directly to the solution.