Problem 84 - Algebra and Trigonometry

Question

\(79-92\) Solve the equation for the indicated variable. $$ a-2[b-3(c-x)]=6 ; \text { for } x $$

Step-by-Step Solution

Answer

The solution for \( x \) is \( \frac{a - 2b + 6c - 6}{6} \).

Step 1: Expand the Brackets

Start by expanding the expression inside the brackets. The expression is \( -2[b-3(c-x)] \). Distribute the \( -2 \) across the terms inside the bracket:\[-2[b-3(c-x)] = -2b + 6(c-x)\]Now distribute the 6:\[6(c-x) = 6c - 6x\]So, the expression becomes \( -2b + 6c - 6x \).

Step 2: Substitute Back Into Original Equation

Substitute the expanded expression from Step 1 back into the original equation:\[a - 2b + 6c - 6x = 6\]

Step 3: Rearrange the Equation

Rearrange the equation to solve for \( x \). Move all terms that do not contain \( x \) to the right side of the equation:\[-6x = 6 - a + 2b - 6c\]

Step 4: Solve for x

Divide every term by \( -6 \) to isolate \( x \):\[x = \frac{-1}{-6}(6 - a + 2b - 6c)\]Simplify the expression:\[x = \frac{a - 2b + 6c - 6}{6}\]

Key Concepts

Isolating Variables

When solving algebraic equations, isolating the variable of interest is a crucial step. This means rearranging the equation so that the variable stands alone on one side. In our exercise, the task is to solve for \( x \) in the equation \[a-2[b-3(c-x)]=6\]The goal is to get \( x \) on one side of the equation by itself. We achieve this by performing a series of algebraic manipulations. Start by expanding and then rearranging the equation step by step. Each operation should simplify the equation progressively until \( x \) is isolated.
To isolate \( x \) in the example, we:

• First eliminate complex expressions using distribution.
• Move other variables and constants to the opposite side using basic arithmetic operations (addition or subtraction).
• Finally, divide or multiply as necessary to solve directly for \( x \).

This careful progression ensures clarity and correct final outcomes.

Distributive Property

The distributive property is a foundational principle in algebra. It allows us to remove parentheses by distributing a factor over terms inside a bracket. In the expression \[-2[b-3(c-x)]\]we apply the distributive property to simplify. We distribute \( -2 \) to each term inside the bracket:

• Multiply \( -2 \) by \( b \), resulting in \( -2b \).
• Multiply \( -2 \) by \( -3(c-x) \), yielding \( 6(c-x) \).

Next, we further distribute inside the inner parentheses to obtain:

• \( 6c \) from \( 6 \times c \).
• \(-6x \) from \( 6 \times -x \).

By consistently applying the distributive property, the equation becomes simpler, paving the way for the solution.

Equation Rearrangement

Equation rearrangement involves changing the arrangement of terms to facilitate isolating a variable. To rearrange effectively, consider using inverse operations and careful arithmetic. Beginning with \[a - 2b + 6c - 6x = 6\]the goal is to move all terms without \( x \) to the right side. This involves:

• Subtracting or adding terms on both sides to keep the equation balanced.
• Moving terms step-by-step: Add \( a \), subtract \( 2b \) and subtract \( 6c \) from both sides.

Thus, \[-6x = 6 - a + 2b - 6c\]Finally, by dividing each term by \(-6\), \( x \) is isolated. Such rearrangement is vital as it simplifies the solving process and leads to the correct solution, i.e., \[x = \frac{a - 2b + 6c - 6}{6}\]Practicing these steps helps develop a systematic approach to solving complex equations.