Problem 11 - Algebra for College Students

Question

Solve each equation. \(0.11 x+0.12(900-x)=104\)

Step-by-Step Solution

Answer

The solution is \(x = 400\).

Step 1: Distribute the 0.12

Expand the equation by distributing 0.12 into the terms within the parentheses: \(0.11x + 0.12(900 - x) = 104\). This yields: \[0.11x + 0.12 \times 900 - 0.12x = 104\] or, \[0.11x + 108 - 0.12x = 104\].

Step 2: Combine like terms

Combine the \(x\) terms on the left side of the equation: \[0.11x - 0.12x = -0.01x\].So the equation becomes: \[-0.01x + 108 = 104\].

Step 3: Isolate the variable term

To isolate the \(-0.01x\) term, subtract 108 from both sides:\[-0.01x + 108 - 108 = 104 - 108\].This simplifies to:\[-0.01x = -4\].

Step 4: Solve for \(x\)

Divide both sides of the equation by \(-0.01\) to solve for \(x\):\[x = \frac{-4}{-0.01}\].Calculating the right side gives:\[x = 400\].

Key Concepts

Distributive Property

The distributive property is a foundational concept in algebra that allows us to multiply a single term by each term within a set of parentheses. This principle simplifies expressions and is particularly useful in solving linear equations.
When you have an equation like \(0.11x + 0.12(900 - x) = 104\), you need to distribute the 0.12 across both terms inside the parentheses: 900 and \(-x\).
In our example:

• Multiply 0.12 by 900, which equals 108.
• Multiply 0.12 by \(-x\), resulting in \(-0.12x\).

Consequently, the expression transforms:
\(0.11x + 0.12 \times 900 - 0.12x\), which simplifies to \(0.11x + 108 - 0.12x\).
Now the equation is ready for the next steps, such as combining like terms.

Combining Like Terms

Combining like terms is a method used to simplify equations by merging terms that have the same variables raised to the same power. This step is crucial for solving equations more easily.
After distributing the terms in our example \(0.11x + 108 - 0.12x = 104\), you notice that there are two terms containing the variable \(x\): \(0.11x\) and \(-0.12x\).
To combine these, subtract \(0.12x\) from \(0.11x\):

• This results in \(-0.01x\).

The equation now looks like \(-0.01x + 108 = 104\).
Combining like terms simplifies the equation further and prepares us for isolating the variable, enabling us to find its specific value.

Isolating Variables

Isolating variables is the process of rearranging an equation to get the variable by itself on one side of the equation. This step is essential for identifying the value of the variable, which completes the equation-solving process.
In the final stages of solving the equation \(-0.01x + 108 = 104\), the goal is to isolate \(-0.01x\).
Begin by subtracting 108 from both sides:

• This gives you \(-0.01x = -4\).

Once you isolate the term with the variable \(x\), solve for \(x\) by dividing both sides by \(-0.01\). This division provides the solution:

• \(x = \frac{-4}{-0.01}\), which simplifies to \(x = 400\).

By isolating the variable, we determine that \(x\) equals 400, successfully solving the equation. This process is key in handling linear equations across various mathematical scenarios.