Problem 68 - Elementary Algebra

Question

Solve. $$ 1-54(4 x-1)=5(12-x) $$

Step-by-Step Solution

Answer

\(x = \frac{-5}{211}\)

Step 1: Distribute the Multiplication

Distribute the numbers outside the parentheses across each term inside the parentheses:\(1 - 54(4x - 1) = 5(12 - x)\)This becomes:\(1 - 216x + 54 = 60 - 5x\)

Step 2: Combine Like Terms

Combine the constant terms on the left side of the equation:\(1 + 54 - 216x = 60 - 5x\)This simplifies to:\(55 - 216x = 60 - 5x\)

Step 3: Move Variable Terms to One Side

Add \(216x\) to both sides to move all terms involving \(x\) to the right side:\(55 = 60 + 211x\)

Step 4: Isolate the Variable

Subtract 60 from both sides to get the \(x\)-term alone:\(55 - 60 = 211x\)This simplifies to:\(-5 = 211x\)

Step 5: Solve for x

Divide both sides by 211 to solve for \(x\):\(x = \frac{-5}{211}\)

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that gives you the power to simplify complex expressions by multiplying a single term with every term inside a set of parentheses. To apply the distributive property, you multiply the outside term by each term inside the parentheses.
For example, in the expression \(54(4x - 1)\), you distribute 54 to both \(4x\) and \(-1\). This results in \(54 \cdot 4x - 54 \cdot 1\). Calculating each multiplication gives us \(216x - 54\).
This property helps you remove parentheses and simplify expressions, making it easier to solve equations.

Combine Like Terms

Combining like terms is an essential skill that helps to simplify algebraic expressions even further. "Like terms" are terms that have identical variables raised to the same power. This means you can only combine terms that have the same variable(s) with the same exponent(s).
For instance, in the expression \(55 - 216x = 60 - 5x\), terms like \(55\) and \(60\) are constants and can be combined separately from those with variables. To simplify, adjust constants together and similarly adjust terms involving the variable \(x\).

• This leaves expressions like \(55\) and \(60\) as possible simplifications, while \(216x\) and \(-5x\) need rearranging to combine properly.

By doing this, you're setting the groundwork for isolating and solving for the variable effectively.

Isolate Variable

Isolating the variable is the process of performing algebraic manipulations to get the variable of interest by itself on one side of the equation. This step is crucial for figuring out what the variable equals.
From our simplified equation \(55 = 60 + 211x\), we must ensure \(x\) stands alone. Begin by removing constants from the side where \(x\) is found. Subtract 60 from both sides, such that you're left with \(55 - 60 = 211x\).
The outcome \(-5 = 211x\) means \(x\) is isolated on the side, preparing us for solving directly. Always keep operations balanced for both sides of the equation, just like a scale remaining level.

Solve for x

Once the variable is isolated, the final step is to solve for \(x\). This involves doing whatever operation remains to make \(x\) equal its solution. With \(-5 = 211x\), you solve for \(x\) by addressing multiplication.
To solve, undo the multiplication by dividing both sides by 211. Here, sincerity in the operation is critical.

• \[x = \frac{-5}{211}\] ensures that the value of \(x\) can be explicitly calculated.

This shows that every step taken up to this point has led to our answer: \(x = \frac{-5}{211}\).
Carefully execute each operation to solve any algebraic equation, always relying on accurate arithmetic principles.