Question

Solve the equation \(-6 x+3(4 x-1)=9 .\) Organize your work into two columns. In the left-hand column show the solution steps. In the right-hand column explain the transformation you used in each step.

Short answer

\(x = 2\)

Step-by-step solution

Expand the brackets using the distributive property

First, use the distributive property for the term \(3(4x - 1)\). Multiply 3 with every term inside the bracket to get \(12x - 3\). The equation becomes \(-6x + 12x -3 = 9\).

Combine like terms

After expanding the bracket, combine like terms. Add \(-6x\) and \(12x\) to get \(6x\), so now the equation becomes \(6x - 3 = 9\).

Isolate the variable

Begin to isolate the variable \(x\) by adding 3 to both sides of the equation, which removes the \(-3\) from the left side. The equation becomes \(6x = 12\).

Solve for x

To solve for \(x\), divide both sides of the equation by 6, which makes the equation become \(x = 2\).

Key concepts

Distributive Property

The distributive property is fundamental in simplifying algebraic equations and it allows us to remove brackets by distributing a multiplier to each term within the bracket. In the given exercise, the distributive property is applied to the term 3(4x - 1).

Imagine you’re sharing 3 sets of items that include 4 x’s and 1 item to be removed. Instead of taking the items out set by set, the distributive property lets you share out the 4 x’s and the single item all at once. In mathematical terms, this means multiplying 3 by both 4x and -1. The result is 12x - 3, which replaces the original term in the equation, eliminating the brackets and simplifying the expression.

Combining Like Terms

Combining like terms is like organizing a group of different objects into piles of the same type. In algebra, it involves adding or subtracting terms that have the same variable raised to the same power. After using the distributive property, our equation had both -6x and 12x, which are like terms because they both contain the variable x to the same power.

To combine them, think of having 6 negative x's and adding 12 positive x's - you'd end up with 6 positive x's left. That's what we do here, simplifying -6x + 12x into 6x. This step is crucial for making the equation more manageable and getting us one step closer to finding the value of x.

Isolate the Variable

Isolating the variable means to get the variable on one side of the equation all by itself. The goal is to have something like x = some number, which tells us the value of x. In our exercise, we needed to isolate x by getting rid of the -3 on the same side as 6x.

Imagine the -3 as a guest that's overstayed their welcome; you want to 'add' enough to it to make it leave, or in this case, balance out to zero. So, we add 3 to both sides to cancel the -3 out. This left us with 6x on one side and 12 on the other, neatly isolating x and setting the stage for the final solution.

Equation Solving Steps

The steps to solve an equation are like a treasure map that guides us to the treasure—the value of the variable. Each step brings us closer to our goal. Our equation began with the distributive property to clear the brackets. Next, we combined like terms to make the equation simpler. We then isolated the variable, making sure it's on its own on one side of the equation. Finally, to find the value of x, we completed the last operation needed: division.

By dividing both sides by the coefficient of x, which was 6, we arrived at x = 2. This value is the 'X' that marks the spot on our map. Our variable x is no longer unknown; we’ve discovered its value through our equation-solving journey, step by step.