Problem 67 - Elementary Algebra

Question

Solve the equation \(2(3 x+1)-5 x=4(x-6)+17\).

Step-by-Step Solution

Answer

Answer: x = 3

Step 1: Distribute the numbers inside the parentheses

Distribute 2 in the expression 2(3x + 1) and 4 in the expression 4(x - 6). \(2(3x+1)-5x = 4(x-6)+17 \Rightarrow 6x+2 - 5x = 4x - 24 + 17\)

Step 2: Combine like terms on both sides of the equation

Combine all the x terms and the constants on both sides of the equation. \((6x-5x) + 2 = (4x) - (24-17) \Rightarrow x + 2 = 4x-7\)

Step 3: Isolate the variable x on one side of the equation

Subtract x from both sides of the equation to move all the x terms to one side. \(x + 2 - x = 4x - 7 - x \Rightarrow 2 = 3x - 7\)

Step 4: Solve for x

Add 7 to both sides of the equation and divide by 3 to find the value of x. \(2 + 7 = 3x - 7 + 7 \Rightarrow 9 = 3x\) Divide both sides by 3: \(\frac{9}{3} = \frac{3x}{3} \Rightarrow 3 = x\) So, the solution is \(x=3\).

Key Concepts

Distributive Property

When faced with an equation like \(2(3x+1)-5x=4(x-6)+17\), the distributive property is your first stop. The distributive property allows you to simplify expressions by eliminating parentheses, making it easier to work with equations. To distribute means to multiply each term inside the parentheses by the number outside.
For example:

• Distribute 2 across \((3x + 1)\), resulting in \(6x + 2\).
• Do the same with 4 for the expression \((x - 6)\), which results in \(4x - 24\).

This step simplifies the structure of the equation, setting the stage for further simplification by combining like terms. Always proceed step by step, ensuring each term is accounted for.
Apply the distributive property carefully, as any oversight can affect the entire solution!

Combining Like Terms

Once you have distributed and rewritten the equation, the next step focuses on combining like terms. This means you'll need to gather similar terms to simplify the equation further. In our example, your equation looks like: \(6x + 2 - 5x = 4x - 24 + 17\).
Here's how combining works:

• Group the \(x\) terms together: \(6x - 5x\) becomes \(x\).
• Combine constant terms on the right side: \(-24 + 17\) simplifies to \(-7\).

Now, you're left with a simpler equation: \(x + 2 = 4x - 7\). Combining like terms reduces the overall number of terms, paving the way for isolating variables in the next step. It's like organizing your notes before starting a project—clear and concise for easier problem-solving.

Isolating Variables

To isolate the variable means to get it alone on one side of the equation. This is a crucial step in solving for the unknown, which in this case is \(x\). From the equation \(x + 2 = 4x - 7\), you'll need to rearrange terms to have all \(x\) terms on one side and constants on the other.
Here's what happens:

• Subtract \(x\) from both sides, simplifying to \(2 = 3x - 7\).
• Next, add 7 to both sides, resulting in \(9 = 3x\).

By strategically subtracting and adding terms, you shift and shuffle the equation to isolate \(x\). This step demands patience and precision. Remember, the goal is to have your variable clear of other terms. Once successfully isolated, solving for \(x\) becomes straightforward.

Algebraic Manipulation

Algebraic manipulation is the final frontier before solving the equation. Once the variable \(x\) is isolated with a coefficient, your task is to remove the coefficient and reveal the variable's value. From the simplified form \(9 = 3x\), division will be your tool.
Executing algebraic manipulation involves:

• Dividing both sides by 3, you simplify \(9 = 3x\) to \(3 = x\).

Here, dividing both sides makes the equation balanced, leaving \(x\) alone on one side. Algebraic manipulation synthesizes all previous steps to find the solution, which, in this case, is \(x = 3\).
This concluding step demonstrates the power of methodical problem-solving: distribute, combine, isolate, and manipulate.