Problem 40 - Elementary Algebra

Question

Solve the equation \(3[3(x-2)+4 x]-24=0\).

Step-by-Step Solution

Answer

Question: Solve the equation \(3[3(x-2)+4x]-24=0\). Answer: \(x = 2\)

Step 1: Distribute 3

We will distribute the \(3\) to both terms inside the brackets: \(3[\underline{3(x-2)}+4 x]-24= 3(3x-6)+4x\times 3 -24\) Now our equation looks like this: \(9x - 18 + 12x - 24 = 0\)

Step 2: Combine like terms

Now, we will combine the \(x\) terms and the constant terms: \((9x + 12x) - (18 + 24) = 0\) This simplifies to: \(21x - 42 = 0\)

Step 3: Isolate the x term

We will add \(42\) to both sides of the equation to isolate the \(x\) term: \(21x - 42 + 42 = 0 + 42\) This simplifies to: \(21x = 42\)

Step 4: Solve for x

Finally, we will divide by \(21\) to find the value of \(x\): \(\frac{21x}{21} = \frac{42}{21}\) This simplifies to: \(x = 2\) So, the solution for the given equation is \(x = 2\).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to simplify expressions by eliminating parentheses. In this exercise, we use it to expand an expression that is multiplied by a number. Specifically, this involves taking a number (in this case, 3) and multiplying it by each term inside the bracket individually.

When you see an equation like \(3[3(x-2) + 4x]\), you need to distribute the \(3\) to both \(3(x-2)\) and \(4x\). Now, it is handled one by one:

• Multiply 3 by \(3(x - 2)\), which results in \(3 \times 3x - 3 \times 6\).
• Multiply 3 by \(4x\), getting \(12x\).

Once these calculations are done, the expression expands to \(9x - 18 + 12x\). By distributing the 3, we simplify the original equation's complexity, making it easier to work with as we continue solving it.

Combining Like Terms

Combining like terms is the next essential step in solving equations. Like terms are terms that contain the same variables raised to the same power. It's crucial to identify and group these terms to further condense an equation.

In our example, once we've distributed and expanded, the equation becomes \(9x - 18 + 12x - 24 = 0\). To combine like terms:

• Look at the variable terms: \(9x\) and \(12x\). Since both contain \(x\), they can be added together to get \(21x\).
• Now, handle the constant terms: \(-18\) and \(-24\). When you add these, you get \(-42\).

After combining, the equation is simplified to \(21x - 42 = 0\). This consolidated expression makes the task of finding the solution to the equation more straightforward, as you're dealing with fewer terms.

Isolating the Variable

Isolating the variable is the key to solving any algebraic equation, as it allows you to solve for the unknown. Our goal is always to have the variable on one side of the equation and everything else on the other.

In the simplified equation \(21x - 42 = 0\), the task is to bring \(x\) by itself on one side. The process involves a few steps:

• First, add 42 to both sides of the equation to cancel out the \(-42\). This will give you \(21x = 42\).
• Next, divide each side by 21 to isolate \(x\). When you divide, you have \(\frac{21x}{21} = \frac{42}{21}\).
• Simplifying the right side results in \(x = 2\).

Now, the variable \(x\) is isolated, and we have found its value: \(x = 2\). Understanding how to systematically isolate the variable helps in solving various algebraic equations.