Problem 21 - Elementary Algebra

Question

Solve each of the equations. $$0.09 x=550-0.11(5400-x)$$

Step-by-Step Solution

Answer

The solution is 2200.

Step 1: Distribute the negative over the parenthesis

Begin by expanding the expression on the right side of the equation. Distribute the \(-0.11\) across \(5400-x\). This results in \(-0.11 \ imes 5400 + 0.11x\).

Step 2: Simplify the expression on both sides

After distributing, the equation becomes: \(0.09x = 550 - 594 + 0.11x\). Simplify these constants to get: \(0.09x = -44 + 0.11x\).

Step 3: Rearrange the equation

Move the terms involving \(x\) to one side by subtracting \(0.11x\) from both sides: \(0.09x - 0.11x = -44\), resulting in \(-0.02x = -44\).

Step 4: Solve for x

Divide both sides of the equation by \(-0.02\) to get \(x = \frac{-44}{-0.02}\). Simplifying this division gives \(x = 2200\).

Key Concepts

The Distributive Property in Action

The distributive property is a fundamental concept in algebra that simplifies expressions and solves equations effectively. Understanding it is crucial for dealing with expressions that include parentheses. In simple terms, the distributive property states that if you have an expression like \(a(b+c)\), it can be expanded to \(ab + ac\). This property ensures that multiplication is distributed over addition or subtraction.
In the given equation, we apply the distributive property to the term \(-0.11(5400-x)\). Here, \(-0.11\) is multiplied by each term inside the parentheses individually.

• First, \(-0.11\) is multiplied by 5400, resulting in \(-0.11 \times 5400\).
• Next, \(-0.11\) is multiplied by \(-x\), resulting in \(0.11x\) due to the multiplication of two negatives producing a positive.

This step transforms the original expression into: \(0.09x = 550 - 594 + 0.11x\). The distributive property helps in breaking down complex expressions, simplifying the process of solving equations.

Simplifying the Equation

Equation simplification is about making an equation easier to handle or solve by combining like terms and reducing clutter. After applying the distributive property in our exercise, we connect constant numbers and variables to simplify.
At this stage, the focus is on streamlining both sides of the equation. The left-side equation \(0.09x = 550 - 594 + 0.11x\) can be simplified by performing the arithmetic for the constants first.

• The subtraction of 550 and 594 results in \(-44\). Thus, the equation now reads \(0.09x = -44 + 0.11x\).

This simplification is about consolidating the constants, which is a vital part of working equations. Simplified equations are much easier to solve, improving both understanding and efficiency during problem-solving.

Isolating the Variable

Isolating the variable is the ultimate goal in solving any algebraic equation. It involves getting the variable on one side of the equation with a coefficient of 1 or itself. In our step-by-step process, we aim to isolate \(x\) in the equation \(0.09x = -44 + 0.11x\).
Firstly, we need to move all terms involving \(x\) to one side of the equation. We achieve this by subtracting \(0.11x\) from both sides:

• This subtraction leads to \(0.09x - 0.11x = -44\), simplifying to \(-0.02x = -44\).

Now, with \(x\) isolated on one side, we solve by dividing both sides by \(-0.02\). Thus, \(x\) is calculated as \(x = \frac{-44}{-0.02}\), which simplifies to \(x = 2200\).
The concept of variable isolation is about manipulating the equation so that solving for the unknown becomes straightforward. This is a core part of learning algebra, as it enables you to find solutions effectively.