Problem 56 - Elementary Algebra

Question

Solve each of the following equations and express the solutions in decimal form. Your calculator might be of some help. $$0.10 x+0.12(720-x)=80$$

Step-by-Step Solution

Answer

The solution is x = 320.

Step 1: Distribute

First, distribute the 0.12 inside the parentheses: \[ 0.10x + 0.12 \times 720 - 0.12 \times x = 80 \]This results in the equation: \[ 0.10x + 86.4 - 0.12x = 80 \]

Step 2: Combine Like Terms

Combine the terms with \(x\): \[ 0.10x - 0.12x + 86.4 = 80 \]This simplifies to: \[ -0.02x + 86.4 = 80 \]

Step 3: Isolate the Variable

Subtract 86.4 from both sides to isolate the term with \(x\): \[ -0.02x = 80 - 86.4 \]\[ -0.02x = -6.4 \]

Step 4: Solve for x

Divide both sides by \(-0.02\) to solve for \(x\): \[ x = \frac{-6.4}{-0.02} \]\[ x = 320 \]

Key Concepts

Distributive Property in Algebra

The distributive property is a key concept that is often used when solving algebraic equations. It involves multiplying a single term by each term inside parentheses. This property helps in eliminating the parentheses to make the equation simpler to solve. In our example, the equation is \( 0.10x + 0.12(720-x) = 80 \). Recognizing that 0.12 needs to be distributed is the first step. Here's how it works:

• Multiply 0.12 by 720, giving us 86.4.
• Then multiply 0.12 by \(-x\), resulting in \(-0.12x\).

After applying the distributive property, the equation converts to \( 0.10x + 86.4 - 0.12x = 80 \). This transformation is crucial because it simplifies the problem, eliminating the brackets and setting the stage for combining like terms.

Combining Like Terms

Once you've applied the distributive property, combining like terms is the next logical step. Combining like terms means bringing similar terms together to simplify the equation further. Terms are considered 'like' if they have the same variable raised to the same power. In our equation, after distribution, the terms with \(x\) are \(0.10x\) and \(-0.12x\). These are like terms because both contain the variable \(x\):

• Combine these like terms to get \(-0.02x\).

After combining, the simplified equation becomes \(-0.02x + 86.4 = 80\). Combining like terms makes the equation easier to manage, setting the foundation for isolating the variable, which is our next crucial step.

Isolating the Variable

Isolating the variable involves manipulating the equation until the unknown, in this case \(x\), is by itself on one side of the equation. This allows us to find the exact value of \(x\). The equation we simplified before was \(-0.02x + 86.4 = 80\). The steps to isolate \(x\) include:

• Subtracting 86.4 from both sides to remove the constant term, leaving us with \(-0.02x = -6.4\).
• Dividing every term by \(-0.02\) to finally solve for \(x\).

This division yields \(x = 320\). Isolating the variable is critical as it lets us express the solution clearly and ensures that our answer is precise and in decimal form. With practice, these steps become intuitive, enabling you to tackle more complex equations with confidence.