Problem 49 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ 2(6-2 x)=-9 x-\frac{1}{2}(-4 x+6) $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{9}{7}\).\

Step 1: Distribute

First, distribute the constants within the parentheses, giving \(2*6 - 2*2x = -9x - \frac{1}{2}*(-4x+6)\) which simplifies further to \(12 - 4x = -9x -2x +3\).

Step 2: Combine Like-Terms

Combine the like-terms on both sides, which simplifies to \(12 - 4x = -11x + 3\). This further simplifies to \(4x - 11x = 3 - 12\).

Step 3: Simplify

This simplifies to \(-7x = -9\). This can be further simplified by dividing each side by -7, to isolate the 'x' on one side. Giving \(x = \frac{9}{7}\).

Key Concepts

Distributive Property

In algebra, the distributive property is essential when working with expressions inside parentheses. The property states that a number outside the parentheses is to be multiplied with every term inside. So, when you see something like \(2(6-2x)\), you apply the distributive property by multiplying 2 with 6 and 2 with \(-2x\). This results in \(12 - 4x\).

This technique helps break down complex expressions, making them easier to work with. In our original exercise, we also needed to apply the distributive property to the term \(-\frac{1}{2}(-4x+6)\), resulting in \(2x - 3\) after distribution. This step is crucial for solving the equation systematically.

Combining Like Terms

Combining like terms is a way to simplify equations by merging terms that have the same variables raised to the same power. This process makes an equation easier to solve because it reduces the number of terms.

For example, in the expression \(12 - 4x = -9x - 2x + 3\), the terms \(-9x\) and \(-2x\) are like terms because both contain the variable \(x\). You can combine these by adding their coefficients, resulting in \(-11x\).

This step simplifies the equation to \(12 - 4x = -11x + 3\), reducing clutter and making the path to the solution clearer. Being proficient at identifying and combining like terms is a crucial skill in algebra.

Solving for a Variable

Solving for a variable means isolating the variable on one side of the equation to find its value. Once you've combined the like terms, you can rearrange the equation to focus on the variable.

In our example, after combining like terms, the equation becomes \(4x - 11x = 3 - 12\). The goal is to have \(x\) by itself on one side. Move all terms involving \(x\) to one side and constant numbers to the other, giving \(-7x = -9\).

The solution requires dividing both sides by \(-7\) to isolate \(x\), resulting in \(x = \frac{9}{7}\). It's important to perform the same operation on both sides of the equation to maintain balance.

Algebraic Simplification

Simplifying an algebraic expression involves making it as concise and readable as possible. Throughout the problem-solving process, you employ techniques like applying the distributive property and combining like terms to make expressions less complex.

In the equation provided, after distributing and combining like terms, we performed simplification by rearranging terms and reducing both sides as much as possible. The expression simplifies from a more complex form into \(-7x = -9\). This step-by-step simplification allows you to see the structure of the problem clearly and guides you to solve for the variable effectively.

Simplification is akin to cleaning up an expression, making it tidy and straightforward, which, in turn, makes further operations much easier and less prone to errors.