Problem 36 - Elementary Algebra

Question

Solve. $$5-2(3 x+1)-1=0$$

Step-by-Step Solution

Answer

\( x = \frac{1}{3} \)

Step 1: Distribute the -2

First, distribute the -2 across the expression inside the parentheses: \(5 - 2(3x + 1) - 1 = 5 - (2 \cdot 3x) - (2 \cdot 1) - 1 = 5 - 6x - 2 - 1\).

Step 2: Simplify the Equation

Combine the constant terms on the left-hand side: \(5 - 2 - 1 = 2\). So the equation becomes: \(-6x + 2 = 0\).

Step 3: Isolate \(6x\)

Subtract 2 from both sides of the equation to isolate the \(-6x\) term: \(-6x + 2 - 2 = 0 - 2\). This simplifies to: \(-6x = -2\).

Step 4: Solve for \(x\)

Divide both sides by -6 to solve for \(x\): \(x = \frac{-2}{-6}\). Simplify the fraction: \(x = \frac{1}{3}\).

Key Concepts

Distribution in Algebra

Distribution in algebra is an important concept that involves multiplying a single term with each term inside a group, represented by parentheses. This is also known as the distributive property. In the exercise, the expression \(-2(3x + 1)\) requires distributing the \(-2\) to both terms inside the parentheses. This results in two separate products: \(-2 \cdot 3x\) and \(-2 \cdot 1\). Here's how it looks:

• Multiply \(-2\) by \(3x\) to get \(-6x\).


• Multiply \(-2\) by \(1\) to get \(-2\).


• Combine these to update the expression: \(-6x - 2\).

Mastering distribution allows you to break down complex expressions into simpler parts, making equations easier to manage. Remember, every term inside the parentheses is multiplied by the outside term.

Isolating Variables

Isolating variables is a crucial step in solving linear equations. It involves getting the variable you are solving for, in this case \(x\), by itself on one side of the equation. This process helps in determining the variable's value. In the example, after distributing, our equation becomes \(-6x + 2 = 0\). To isolate \(-6x\), you need to perform operations that eliminate other terms.
Start by subtracting \(2\) from both sides to remove the constant term on the left:

• Subtract \(2\) from the left side: \(-6x + 2 - 2 = -6x\).


• Apply the same subtraction to the right side: \(0 - 2 = -2\).

Now, the equation is simplified to \(-6x = -2\). Isolating the variable is a foundation for solving equations, reinforcing the concept of maintaining balance across both sides of an equation.

Simplifying Expressions

Simplifying expressions is a process of reducing an equation to its simplest form. This often involves basic arithmetic operations like addition, subtraction, and combining like terms. In the provided exercise, after distribution, we deal with the expression \(5 - 6x - 2 - 1\). The goal is to combine all the constant numbers:

• Combine \(5 - 2 - 1\) to get \(2\).

The equation now reads as \(-6x + 2 = 0\). By simplifying early in problem solving, it alleviates the complexity of equations, allowing easier manipulation and understanding. Once simplified, further steps like isolating variables become more straightforward, as you deal with fewer and more manageable terms. Proficiency in simplifying expressions leads to efficiency and accuracy in algebraic problem-solving.